Effect of On-Axis Dip in Refractive-Index Profile on the Cutoff Frequency of Modes of Few Mode Fibers: A Numerical Approach

Abstract

The normalized cutoff frequencies of the linearly polarized (LP) modes of a graded-index few mode fiber have been calculated as a function of profile shape parameters using a simple numerical method. The effect of refractive-index dip on these cutoff frequencies and on the order of appearance of these modes has been investigated. The presence of dip causes an increase in the cutoff V values of all the modes, and the maximum shift in the cutoff V value is found to be even 20% for some modes. The order of appearance of some modes also gets drastically affected.     

Natural frequencies and vibration modes of laminated composite plates reinforced with arbitrary curvilinear fiber shape paths

The development of tow-placement technology has made it possible to control fiber tows individually and place fibers in curvilinear distinct paths in each layer of a laminated plate. This paper presents an analytical method for determining natural frequencies and vibration modes of laminated plates having such curvilinear reinforcing fibers. Spline functions are employed to represent arbitrarily shaped fibers, and Ritz solutions are used to derive frequency equations using series type shape functions. The strain energy is evaluated by numerical integration involving the fiber orientation angle, and is calculated using the derivative of the spline function in minute intervals. The results show that the natural frequencies obtained by the present method agree well with results from finite element analyses. The vibration mode shape contour plots of the plates are seen to reflect clear influences of the fiber shapes.     

Experimental investigation of cutoff phenomena in nonlinear photonic crystal fibers

Modal cutoff is investigated experimentally in a series of high-quality nonlinear photonic crystal fibers. We demonstrate a suitable measurement technique with which to determine the cutoff wavelength and verify the technique by inspecting the near field of the modes that may be excited below and above the cutoff. We observe a double-peak structure in the cutoff spectra, which is attributed to splitting of the higher-order modes. The cutoff is measured for seven different fiber geometries with different pitches and relative hole sizes, and very good agreement with recent theoretical work is found.     

Propagation Characteristics of an Elliptically-cored Highly Birefringent Fiber using an Analytical Method

By using the method of separation of variables in the elliptical coordinate system, a representative formula in terms of Mathieu and modified Mathieu functions is derived and longitudinal field components for the even and odd modes are presented for the proposed fiber. The eigen value equation for the fiber to be considered is obtained by solving the fourth order determinantal equation under weak guidance condition and is given in this paper. The cutoff frequency of several lower orders modes for different eccentricities have been obtained and their propagation characteristics have been plotted. The effect of the eccentricity on the modal cutoff values and mode transmissions is discussed. Finally, the modal birefringence in the proposed fiber is also estimated.     

W型单模单偏振光纤隧道效应的模场分析

给出了W型单模单偏振光纤基模模场在截止前,截止时和截止后的具体计算方法,利用基模模场分析了W型单模单偏振光纤隧道效应发生的过程,强弱和方向,得出了W型单模单偏振光纤隧道效应随其基模的截止而显著增强,并且主要集中在椭圆内包层短半轴方向的结论。     

Enhancement of single mode operation in coaxial optical waveguide using DB boundary conditions

In this study, a competent numerical strategy to compute the dispersion of optical waveguides is presented and propagation of electromagnetic waves in a coaxial optical waveguide with DB boundary conditions is instigated. For this intend, cylindrical coordinates are here being used to derive the DB boundary conditions and to obtain field components for the modes. The propagation constant for the waveguide to be studied is determined by solving the Bessel and the modified Bessel functions. The cutoff frequencies for various lower order modes have been calculated and their dispersion characteristics are plotted correspondingly. The behavior of the coaxial optical waveguide under DB boundary conditions is shown to be significantly different from that of coaxial optical waveguide and conventional optical waveguide under traditional or tangential boundary conditions. Finally, the effect of waveguide dimensions on the mode cutoff frequencies and fabrication issues are also addressed.     

弱渐变抛物型光纤传导模式的截止频率与介电比

在弱渐变假定下,研究了抛物型渐变光纤传导模式的截止频率对折射率参量的依赖关系。采用矢量场法,得到了在弱导条件下传导模式的特征方程和截止频率的近似计算公式。引入了介电比γ,它是由光纤三个折射率参量所确定的常数。在弱导近似的条件下,归一化截止频率仅与γ这一个变量有关,任一渐变抛物型光纤的实际截止频率可通过γ而由归一化截止频率简单求得,从而找到了渐变光纤的实际截止频率与归一化截止频率间的联系。给出了归一化截止频率随γ的变化曲线,归一化截止频率随γ的增加而单调上升,但HElm模截止频率的上升规律与其他模式不同。     

大芯径大数值孔径阶跃光纤传导模特征方程求解

导出了相邻阶第二类变型贝塞尔函数的比值在小宗量的逼近形式,利用Lentz—Thompson方法计算相邻阶第一类贝塞尔函数的比值在小宗量的值,计算精度为浮点数系统的最低有效位。从而解决了求解大芯径、大数值孔径阶跃光纤的特征方程时．贝塞尔函数溢出双精度浮点数表示范围的问题。分别对多模石英光纤和大芯径大数值孔径的聚合物光纤的传导模特征方程进行了求解,石英光纤的传导模特征值计算结果与Optiwave公司的软件一致;对于聚合物光纤,算法给出了所有模式的计算结果,其中模式角向序数小于70的计算结果与Optiwave公司的软件一致,而Optiwave公司的软件不能计算角向序数大于70的模式。     

Slowing down of solitons by intrapulse Raman scattering in fibers with frequency cutoff

A method for transforming fast solitons into slow ones in bandgap fibers is proposed. The approach is based on the deceleration of the solitons by intrapulse Raman scattering, which can be achieved for fiber modes having a cutoff frequency. We develop a comprehensive theory for the soliton slowdown and elucidate how the fiber losses introduce a fundamental minimum for the soliton velocity.     

Interesting effect of optical Kerr nonlinearity in expanding single-mode regime of optical fibers using dispersion-flattened profiles

An interesting effect of optical Kerr nonlinearity is reported from theoretical investigation on the fundamental and first higher mode cutoff frequencies in single-mode fibers doped with highly nonlinear materials like CdS_xSe_1−x with W-type dispersion-flattened profiles. It is observed that whereas the single-mode regime decreases absolutely for dispersion-shifted profiles due to reduction of the first higher order mode cutoff frequency as is evident from earlier studies, it increases for W-type profiles for such nonlinearity taking its effect on the nonzero fundamental mode cutoff frequency into consideration. Thus, one can go further down below the fundamental mode cutoff frequency while using W-fibers in single mode nonlinear fiber devices involving evanescent field coupling in the low V region.     

Cutoff frequencies in planar optical waveguides with arbitrary index profiles: An efficient numerical method

A numerical method for obtaining mode cutoffs for planar waveguides with arbitrary index profiles is developed. The method is based on the Galerkin method in which the wave equation for modes at cutoff is converted to a matrix eigenvalue equation using a set of orthogonal basis functions. Due to different boundary conditions, we have identified two separate cases; one, in which the cover and the substrate indices are identical leading to same behavior of the field at cutoff in these two regions and, the other, in which the two indices are different and hence, the field behaves differently. We have accordingly chosen different basis functions for the two cases. The method results in a generalized matrix eigenvalue problem which has been converted to a standard symmetric matrix eigenvalue analytically. The method has been used to obtain mode cutoffs for waveguides with a variety of index profiles. Comparisons with available exact results show that very good accuracies can be obtained with moderate matrix sizes.     

声辐射模态和球谐波函数在球源声场重建中的比较

声辐射模态和球谐波函数在球源声场重建中均有应用,两者具有很多相似性,都具有相同的多极子辐射模式和"群组"分布特性。有观点认为球形声源的声辐射模态向量即为球谐波函数,但两者的内在关系一直缺乏有效的证明。为分析两种基函数的相似性及在球源声场重建领域的差异,在理论上建立了两者的内在联系,在应用上比较研究了基于两种基函数的声场重建方法的有效性和可靠性。数值算例结果表明,在中高频段时基于两种函数的声场重建效果相当,但在低频段尤其是存在干扰源的非自由场条件下的声场重建应用中,基于声辐射模态的声场重建方法能得到更高的重建精度。可见,声辐射模态具有较为明显的声场重建优越性,这为利用声辐射模态和球谐波函数进行球源声场重建提供了理论参考。      

Mode classification and loss mechanism in air-core Bragg fibers

With an equivalent mode-solving model, the mode spectra in air-core Bragg fibers are systematically studied by using an improved full-vector finite-difference method. All supported modes are classified into four categories, namely, guided modes, cladding modes, leaky modes and PML modes, among which the leaky modes can be further subdivided into radiation-like and evanescence-like leaky modes. To ensure that the modes are solved accurately and efficiently with this model, the strategy for choosing model parameters is suggested. Benefiting from this convenient mode solver, the characteristics of cladding modes are observed in details, and potential applications are suggested. Moreover, the rapid loss of a hybrid mode at lower frequencies is explained by an evanescence-like leaky mode induced cutoff, which is different from that of TE₀₁ mode due to the failure of band gap confinement.     

Guided and surface modes in chiral nihility fiber

Guided and surface modes in the chiral nihility fiber consisting of a chiral nihility metamaterial core with an achiral cladding have been investigated. The dispersion equations of guided and surface modes in the chiral nihility fiber and their cutoff frequencies are derived. Dispersion curves of guided and surface modes are presented and the effect of chirality parameter on dispersion curves is discussed in detail. Some novel propagation characteristics are found.     

Analysis of air-guiding photonic bandgap fibers

We present what is to our knowledge the first theoretical analysis of air-guiding photonic bandgap fibers. The fibers are characterized by a large hollow core and a microstructured cladding exhibiting photonic bandgap effects. Using an efficient, full-vectorial numerical method, we explain the operational principle of the fibers and obtain detailed information about the properties of the air-guided modes. This information includes accurate determination of the modes' spectral extent, cutoff properties, and mode-field distributions.     

Consistency of blocking transformations in the finite-temperature renormalization group

The finite-temperature renormalization group is formulated via the Wilson-Kadanoff blocking transformation. Momentum modes and the Matsubara frequencies are coupled by constraints from a smearing function which plays the role of an infrared cutoff regulator. Using the scalar λφ⁴ theory as an example, we consider four general types of smearing functions and show that, to zeroth order in the derivative expansion, they yield qualitatively the same temperature dependence of the running constants and the same critical exponents within numerical accuracy.     

高功率光纤中传输光模式与其波长相关性研究

针对高功率光纤激光模式诊断和光谱诊断的需求,研究了光纤中传输模式的波长相关性,数值计算了光纤中各个模式的模场分布随波长的变化曲线及相应的光束质量,采用双傅里叶变换F~2法实际测量了光纤模式成分与波长的关系曲线.结果表明,光纤中各个模式的模场分布随波长变化,波长越长,模场面积越大;模式的光束质量随波长变化不大,但在截止频率附近明显变差;光纤中各个模式的功率占比与波长有关.     

三包层WⅠ和WⅡ型单模光纤波导色散特性的研究

研究了三包层WⅠ和WⅡ型单模光纤的波导色散特性。结果发现在相同条件下，三包层WⅠ和WⅡ型单模光纤零色散点的调节范围比传统的双包层W型单模光纤明显增大。详细分析了几何参量P、Q和光学参量R1、R2对单模传输时的波导色散特性和低次模截止频率的影响。所得的研究结果为获得更为理想的色散补偿、色散平坦光纤及设计新型无源光器件提供了重要的依据。计算波导色散的方法可推广到多包层光纤。     

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.     

Prediction and measurement of nonpropagating Lamb modes at the free end of a plate when the fundamental antisymmetric mode A0 is incident

Reflection of Lamb waves when the fundamental mode A0 is incident at the free end of a plate is studied, in order to identify the extent to which the generation of nonpropagating modes influences the field local to the end of the plate. Semi-analytical predictions, finite element simulations, and experimental measurements are presented for frequencies below the A2 cutoff. First it is shown, for frequencies below the A1 cutoff, that reflection of the A0 mode is accompanied by a delay in phase, and that there is significant additional motion due to nonpropagating modes within about five plate thicknesses of the end. The extend of this additional motion in the vicinity of the end of the plate is demonstrated by subtracting the contribution of the propagating modes from the displacement field. The wave field at frequencies above the A1 cutoff is more complex because the A1 as well as the A0 propagating modes are present at the end of the plate. Nevertheless, it has still been possible, using semi-analytical predictions and finite element simulations, to demonstrate the additional motion due to the nonpropagating modes.