Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

Propagation of p-polarized waves in a linearly graded index film surrounded by negative index materials

The theory of step-index waveguides is well-established. Most practical slab waveguide structures have a graded-index profile. The basic properties of graded-index planar waveguide structures are similar to those of step-index waveguides with subtle differences. The most common types of graded-index slab waveguides are linearly and exponentially graded-index profiles. We here treat linearly graded-index slab waveguide. In this work, a three-layer waveguide structure with linearly graded-index film is considered. We assume three structures: the first structure comprises a left-handed material (LHM) cladding, the second structure contains a LHM substrate layer and the third has a LHM cladding and substrate. Closed-form expressions for electric and magnetic fields and the characteristic equation are derived. The three normalized parameters: the asymmetry coefficient (a), the normalized film thickness (V) and the normalized guide index (b) are used to study the dispersion properties of the proposed slab waveguide structure.     

Uniform Core Field in Symmetrical Planar Waveguides and Circular Fibers with Nonlinear Claddings

It is known that no uniform electric field profile exists in a planar linear waveguide. The uniform core field can be shown to exist in symmetrical planar waveguides and circular fibers with nonlinear claddings. Theoretical analysis and numerical calculations are carried out to show that when the modal index equals to the core refractive-index, the core field becomes uniform at an appropriate optical power. Analysis for a step-index circular fiber with nonlinear cladding have also shown that the core field becomes uniform under similar conditions. The occurrence of a uniform field in a waveguide core may believe to have promising applications in waveguide, optoelectronic and photonic devices.     

Loss calculations in bent multimode optical waveguides

We present a straightforward geometric optics method to calculate the power attenuation induced by a circular bend in the axis of a multimode optical waveguide. The method involves tracing of rays and use of generalized power transmission coefficients. Both slab and fibre waveguides (with either step-index or parabolic core profiles) have been considered; the source has been assumed to be Lambertian.     

Analysis of circular fibers with an arbitrary index profile by the Galerkin method

We propose a full-vectorial Galerkin method for the analysis of circular symmetric fibers with arbitrary index profiles. A set of orthogonal Laguerre-Gauss functions is used to calculate the dispersion relation and mode fields of TE and TM modes. Examples are given for both standard step-index fibers and Bragg fibers. For standard step-index fiber with low or high index contrast, the Galerkin method agrees well with the analytical results. In the case of the TE mode of a Bragg fiber it agrees well with the asymptotic results.     

Bragg diffraction of light in a restricted homogeneous system of tunnel-coupled waveguides

A system of tunnel-coupled rectilinear waveguides is studied. The dependence of the number of modes in this system on the number of waveguides N, the distance between the waveguides, and the number of modes in a separate waveguide is considered. It is shown that the modes of the m = N and N + 1 orders in such a system are Bragg modes; i.e., the angle between the direction of their propagation and the system axis is close to the Bragg angle. The effective refractive indices n ^* of these modes change stepwise. The step size Δn ^* is found to be dependent on the distance s between the waveguides and on the number of modes in a separate waveguide. A system of single-mode waveguides with the number of guided modes M = 34 < N = 50 is studied experimentally. It is shown that the Bragg modes of the system lie among the leakage modes of the system and have rather low losses. It is demonstrated that the localization of the Bragg modes among the leakage modes may be favorable for their selection upon light generation.     

非线性光波导中基波和二次谐波的模场分布研究

非线性波导是一种重要的光量子器件。采用马卡梯里近似,可以近似描述波导的模场分布特征。针对ee-e准相位匹配的周期极化铌酸锂波导,分析波长为1 064 nm的基波和532 nm的二次谐波的基模模场分布。结果表明:如果波导为嵌入型正方形波导,基波和二次谐波的光斑基本为圆形,基波场分布范围较大;在截面正方形边长为5 μm并且芯区比包层折射率略高0.02时,波导就能将两种光波场基本约束在芯区中,但是这种波导边长为0.5 μm时约束能力就变得很差;如果采用芯区与包层折射率差别较大的波导,芯区边长为0.5 μm就可以将两种场分布基本约束在芯区中。结果对分析其他类型的非线性波导的模场分布具有指导意义。     

矩形波导与圆柱波导或圆柱谐振腔间的小孔耦合

本文讨论了三种不同系统的小孔耦合的近似解法:矩形波导与圆柱波导的同轴耦合,矩形波导中的TE₁₀型波与圆柱谐振腔中的TM₁₂₀型波间的耦合和矩形波导中的TE₁₀型波与圆柱波导中的TM_mn型波间的耦合。通过这几个问题的解法,我们希望读者可以看到处理复杂微波元件问题时所需的从物理概念出发的灵活应用已有方法的必要性。这里提供的几个公式有着广泛的应用范围而是在现有文献中探讨得不够的。这里介绍的方法可以应用到其他新的微波网络问题中去。     

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.     

Perturbation analysis of finitely clad optical waveguides and couplers

In calculating the modal characteristics of a finitely clad optical waveguide, if the wavelength is much shorter than the core of the waveguide, then the cladding of the waveguide can be assumed to be infinite. On the other hand, if the wavelength is much longer than the core, the core can be ignored. These two approximations, although simple and accurate in the two respective extreme cases, do not provide adequate information in the transition region, where many tapered devices are operated. To overcome this a novel perturbation method is devised, in which these two approximations are combined in such a way that a result that is accurate in all cases can be obtained. Analytical expressions based on this method for describing the dispersion characteristics of finitely clad slab waveguides and circular step-index fibres are derived and compared with other conventional approximations. The combined use of this method and the coupled-mode theory for the analysis of optical couplers is also described.     

Spectrally selective splitters with metal-dielectric-metal surface plasmon waveguides

Spectrally selective splitters with metal-dielectric-metal (MDM) subwavelength waveguides are proposed in this paper. The method is based on the Bragg grating structure with periodically modulated MDM waveguide width, which delivers a stop band effect for the surface plasmon propagating in the waveguide. By adding appropriate Bragg grating structure in one or more arms of the waveguide splitters, light in a certain frequency range can be readily guided into the desired directions, as demonstrated numerically by the finite-difference time-domain method. Dependence and optimization of the geometrical parameters are also considered in this paper.     

Transverse mode competition in defect states of a waveguide with corrugated walls

We investigate the defect states arising in the Bragg and non-Bragg gaps by inserting a straight duct into a waveguide with periodically corrugated walls. In periodic waveguides, the Bragg gap is created by the interference of the same transverse modes whereas the different mode coupling leads to the non-Bragg one. Due to the involved high-order modes, there are two defect states observed in the non-Bragg gap while only one in the Bragg gap, indicating that transverse modes play a significant role in the creation of defect states. Furthermore, the frequency of each defect state highly relies on the defect geometries and their band widths can be optimized by the number of waveguide segments. The proposed transverse mode competition analysis reveals the mechanism of frequency shifting and provides an opportunity for guided wave control engineering, which would definitely benefit their applications in various functional devices, such as filters, sensors, and amplifiers.     

SHG in Bragg reflection waveguides for enhanced bandwidth applications

A novel scheme for second harmonic generation in asymmetric Bragg reflection waveguides is presented for enhancing the bandwidth of non-linear interaction. The strong dispersion exhibited by the Bragg reflection waveguide is exploited for countering the rapidly falling efficiency as one moves away from the phase-matching wavelength. The role of the periodic cladding on the efficiency and bandwidth of second harmonic generation in a lithium niobate based Bragg reflection waveguide is analyzed. Such an idea could be used in other parametric processes for enhancing the bandwidth and for easing fabrication tolerances.     

Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide

A new type of wavelength selective filter, based on high differential dispersion between two coupled waveguides, is presented. The Bragg Reflection Waveguide displays high effective refractive index dispersion, due to the interaction of the guided mode with the two confining Bragg reflectors. When coupled with a weakly guided buried channel silica waveguide, a very narrow bandwidth filter (<1 nm) can be easily produced, in a shorter length, with respect to directional couplers made with standard step index channel waveguides. The complete design methodology, fabrication and characterization are presented.     

Improved photonic crystal directional coupler with short length

We investigate a photonic crystal (PC) waveguide coupler which is formed by two closely spaced linear waveguides in a two-dimensional triangular lattice of air holes. Our study shows that shifting one row of the air holes between the waveguides affects the dispersion curves of the guided modes and if the triangular lattice of air holes between the waveguides is replaced by a rectangular lattice, this modification results in an ultra-short coupling structure with coupling length less than 3a, where a is the lattice constant. Also, we investigate the effect of changing the radii of air holes that are adjacent to or between the waveguides on the coupling length and show that increasing the radius of air holes between the waveguides decreases the coupling length. We analyze the output spectrum of an ultra-short channel drop filter designed based on this structure.     

Numerical Method of Determining the Cutoff Frequency and Mode Speckle Pattern of Optical Waveguides

Based on a numerical solution of the nonstationary wave equation, an algorithm for determining the eigenfunctions and eigenvalues of the LP_im modes of axially symmetric optical waveguides is suggested. The cutoff frequency of the LP₁₁ mode of different waveguides is determined using the algorithm constructed. The results obtained are compared with the well-known analytical formulas and results of numerical calculations for the equivalent step-index waveguide model. Based on the analytical results obtained, an eigenfunction and eigenvalue search algorithm is suggested for a waveguide with a two-dimensional refractive index profile (in particular, for an axially nonsymmetric waveguide).__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 11–16, January, 2005.     

缺陷态同轴布拉格波导传输特性

采用基于Ansoft HFSS的电磁仿真平台,研究了一个工作于Ka波段（37.5 GHz附近）的缺陷态同轴布拉格波导在不同的开槽方式及缺陷位置条件下的传输特性。结果表明:该波导只有在内外导体壁单面开槽并引入缺陷,或者双面开槽且两个开槽面同时引入缺陷的条件下才能够有效地产生出传输通带;对于后者而言,将内外导体壁的波纹初始相位差设置为且让缺陷位于波导中间时,可以获得带宽极窄、阻带抑制范围较宽的传输通带,此时,给内外导体壁的缺陷引入适度的错位能够对通带的中心频率、带宽等性能指标起到微调作用,而引入大间距的错位则能够在禁带中产生出多个通带,从而起到多通道滤波的作用。     

Effect of Distortions on an Annular Optical Fiber Waveguide

We present the modal analysis and dispersion curves of some asymmetric annular optical-fiber waveguides. The proposed cross-sections can be conceived as having the shape of a distorted annular circular waveguide and leads to a tolerance study of deformation in an annular optical-fiber waveguide. We obtain mathematical expressions for the modes of the proposed waveguides using boundary matching techniques under some logical assumptions. We observe that the proposed distortions are able to tune the cutoff values of annular circular waveguide at desired levels.     

Surface electromagnetic waves at the interface between a homogeneous medium and a system of coupled waveguides

An analysis is made of the experimental investigations of the surface waves existing at the interface between a homogeneous medium and a periodically stratified medium that represents a bounded system of coupled waveguides. It is shown that, in all cases of the observation of surface waves, the bounded system of coupled waveguides has its own spectrum of guided modes and a spectrum of leaky modes that become surface waves of the system. It is also demonstrated that a biosensor can successfully operate when two surface waves serve as leaky modes of a Bragg waveguide in which the periodic system of waveguides is used as a distributed Bragg mirror of this waveguide. A structure supporting surface waves is designed on the basis of ten pairs of Nb₂O₅-SiO₂ layers and implemented experimentally. The surface waves are detected with a K8 glass prism according to the Kretschmann scheme.     

Femtosecond laser direct writing of multiwavelength Bragg grating waveguides in glass

Novel Bragg grating waveguide structures have been fabricated in bulk borosilicate glass through a type II photosensitivity mechanism driven by single femtosecond laser pulses. Low-loss single-mode waveguides and narrow-linewidth Bragg gratings were generated simultaneously by forming an array of refractive index voxels in a single laser scan. Laser pulse duration is shown to significantly affect the grating strength and waveguide loss. Bragg wavelengths, defined by the periodicity of laser-modified volumes, were fully controlled by the sample scan speed to provide resonances anywhere in the 1200-1620 nm telecommunication bands. Four linear Bragg filters with distinct resonant wavelengths are presented that define the first demonstration of laser writing of multiple-wavelength and cascaded Bragg grating waveguides in a single process step.