Single mode operation of elliptical gain guided and index antiguided fibers

Elliptical gain guiding fibers in which gain guiding effects are dominant compared to conventional index guiding fiber is analyzed by solving Mathieu equations with complex-valued fiber parameters. The properties of mode propagation and single mode operation are evaluated in this elliptical gain guiding fiber with the assumption of a uniform gain distribution in the active core medium. Threshold for lossless mode propagation increases exponentially with the eccentricity of the elliptical cross section. Further, the difference in threshold between the lowest two order modes is constant for arbitrary eccentricity.     

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.     

Dispersion relation for space-charge waves in a warm plasma-filled elliptical waveguide in an infinite axial magnetic field

A perfectly conducting elliptical cylinder filled with a warm plasma and immersed in an infinite axial magnetic field is considered. Using Maxwell’s equations and dielectric tensor, a Mathieu differential equation for axial component of electric field is obtained. Considering the boundary conditions, dispersion relation for waves in a plasma of warm electrons and immobile ions, which fills an elliptical waveguide and it is under the action of infinite axial magnetic field are calculated. Furthermore, dispersion relation and scalar potential in the quasi-static approximation in a cold magnetized plasma elliptical waveguide is calculated. The obtained results are graphically presented.     

基于轴棱锥产生近似无衍射Mathieu光束的新方法

提出了一种基于轴棱锥产生零阶近似无衍射Mathieu光束的新方法,利用轴棱锥聚焦具有椭圆高斯振幅调制的平面波,得到近似零阶无衍射Mathieu光束.根据椭圆高斯平面波经轴棱锥衍射的衍射积分公式,对光强分布进行了数值模拟,依据几何光学模型计算了近似无衍射Mathieu光束的最大无衍射距离,并设计了实验对理论模拟的结果进行了验证.实验采用柱透镜和准直扩束系统变换圆高斯光束产生具有椭圆高斯振幅调制的平面波,用轴棱锥聚焦该平面波后得到近似无衍射Mathieu光束,实验结果与理论模拟和计算相符.     

椭圆芯手征光纤的基模特性研究

对椭圆芯手征光纤进行了严格的求解，得到了模式场的解析解；推了了模式特征方程；并对基模的偶模和奇模特征方程进行数值求解，研究了手征参数对不同椭圆比下基模的色散曲线，双折射，模间色散等特性的影响。     

Computing Eigen value equation and modal dispersion characteristics of an elliptical Bragg waveguide

Using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a new type of Bragg waveguide having elliptical core cross section with multilayered cladding is derived. The eigen equation is written in the form of Mathieu and the modified Mathieu functions and a dispersion relation is obtained for various modes supported by the proposed Bragg waveguide. The cutoff frequencies for several lower order even-odd modes have been calculated and their propagation characteristics are plotted. The results show that the dispersion curves are discontinuous and modes can exist only in particular wavelength bands. The effects of elliptical eccentricity on the mode cutoff values and mode transmission are addressed. Finally, the modal birefringence in the said waveguide is also estimated.     

椭圆傅里叶级数展开法和椭圆光波导的截止频率

与严格马丢（Ｍａｔｈｉｅｕ）函数、有限元法等许多典型方法及一些实验测试的比较表明,本文提出的计算弱导椭圆光波导的椭圆傅里叶级数展开法具有通用、准确、高效、简便的优点。     

Low eccentricity elliptical fibers with helical windings under slow-wave consideration—Some special cases

Analytical investigation of the lightwave propagation through dielectric elliptical fibers with conducting helical windings on the core-clad boundary is presented under the consideration of slow-wave approximation. The analysis essentially requires the use of elliptical coordinate system. Mathieu and the modified Mathieu functions are used as the representatives of the electromagnetic fields within the structure. Field components in the different sections of the guide are presented, and the characteristic dispersion relations for the system are deduced for 0° and 90° pitch angles. The investigation reveals that the pitch angle has ultimately no effect on the dispersion relations for their selected values. The effect of core ellipse eccentricity on the dispersion characteristics is presented for two different operating wavelengths, and considering different values of the core major axis length.     

Sound propagation in acoustically lined elliptical ducts

Sound propagation in ducts with elliptical cross-sections can be described in terms of modes characterized by Mathieu functions of orders specified by the boundary conditions. For ducts with locally reacting liners there is coupling between modes because the admissible solutions are linear combinations of Mathieu functions of different orders and the eigenvalues are roots of an infinite determinant. The amount of mode coupling depends on the eccentricity of the duct. For the case of small eccentricity of the duct, approximate general solutions are derived and an example is discussed, where solutions are found.     

中性粒子在Ioffe阱中的经典运动方程的Floquet解

研究中性粒子在Ioffe阱中近原点区域的囚禁时,阱中的磁场可以呈现出一种简明的形式.磁矩μ反平行于磁场的中性粒子在阱中与磁场发生相互作用,借助相互作用势,可以获得粒子在阱中的经典运动方程.在一定的条件下,采用迭代近似的目的,将方程演化为马丢方程的形式,利用传统的WKBJ目的可实现方程的近似求解.研究阱中中性粒子的囚禁问题时,感兴趣的是马丢方程的Floquet解,即周期为π,2π的全周期和半周期解,欲获得这种周期解,马丢方程中的参数λ和q必须满足一定的关系,为此必须选择阱的特定参数和粒子的特定初始条件,对这一问题进行了探索性的研究.     

Acoustic scattering by multiple elliptical cylinders using collocation multipole method

This paper presents the collocation multipole method for the acoustic scattering induced by multiple elliptical cylinders subjected to an incident plane sound wave. To satisfy the Helmholtz equation in the elliptical coordinate system, the scattered acoustic field is formulated in terms of angular and radial Mathieu functions which also satisfy the radiation condition at infinity. The sound-soft or sound-hard boundary condition is satisfied by uniformly collocating points on the boundaries. For the sound-hard or Neumann conditions, the normal derivative of the acoustic pressure is determined by using the appropriate directional derivative without requiring the addition theorem of Mathieu functions. By truncating the multipole expansion, a finite linear algebraic system is derived and the scattered field can then be determined according to the given incident acoustic wave. Once the total field is calculated as the sum of the incident field and the scattered field, the near field acoustic pressure along the scatterers and the far field scattering pattern can be determined. For the acoustic scattering of one elliptical cylinder, the proposed results match well with the analytical solutions. The proposed scattered fields induced by two and three elliptical–cylindrical scatterers are critically compared with those provided by the boundary element method to validate the present method. Finally, the effects of the convexity of an elliptical scatterer, the separation between scatterers and the incident wave number and angle on the acoustic scattering are investigated.     

椭圆截面的无限长金属棒对任意平面电磁波在有损耗的均匀各向同性无限媒质中的散射

本文利用椭圆柱形坐标系,严格地分析了具有任意偏心度的椭圆截面的无限长理想导电的金属棒,放在有损耗的均匀各向同性的无限介质中,对从任意方向入射的均匀平面波的散射问题;并将散射场表示为马许函数。     

Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.     

The effect of nonlinearity on unstable zones of Mathieu equation

Mathieu equation is a well-known ordinary differential equation in which the excitation term appears as the non-constant coefficient. The mathematical modelling of many dynamic systems leads to Mathieu equation. The determination of the locus of unstable zone is important for the control of dynamic systems. In this paper, the stable and unstable regions of Mathieu equation are determined for three cases of linear and nonlinear equations using the homotopy perturbation method. The effect of nonlinearity is examined in the unstable zone. The results show that the transition curves of linear Mathieu equation depend on the frequency of the excitation term. However, for nonlinear equations, the curves depend also on initial conditions. In addition, increasing the amplitude of response leads to an increase in the unstable zone.     

Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method

This paper presents a semi-analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries. To satisfy the Helmholtz equation in the elliptical coordinate system, the multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions. The boundary conditions are satisfied by uniformly collocating points on the boundaries. The acoustic pressure at each point is directly calculated in each elliptical coordinate system. In different coordinate systems, the normal derivative of the acoustic pressure is calculated by using the appropriate directional derivative, an alternative to the addition theorem. By truncating the multipole expansion, a finite linear algebraic system is derived. The direct searching approach is employed to determine the natural frequencies by using the singular value decomposition (SVD). Numerical results are widely discussed for several examples including an elliptical cavity, a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders. The accuracy and numerical convergence of the presented method is validated by comparison with available results from the analytical method and the commercial finite-element code ABAQUS. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the key features of the present method thanks to its semi-analytical feature.     

Wave numbers of partially filled rectangular waveguide

An approximate method is proposed for determination of the wave numbers of a partially filled rec angular waveguide. The method employs the eigenvalues and parameters of Mathieu equations. The natural H-and E-waves of a partially filled rectangular waveguide are expressed in terms of Mathieu functions. Numerical results of waven number studies are presented and interpreted graphically.     

Slow-Wave Characteristics of Elliptical Corrugated Waveguides with a Concentric Circular Hole

We present the formulation of elliptical corrugated waveguides with a concentric circular hole using the fieldmatching method and the addition theorem for Mathieu functions. The dispersion equation and the mean interaction impedance of this structure are derived separately. The numerical results, which are generally based on the current approach, agree well with the results obtained by the commercial software package CST. As a slow-wave structure, this structure has potential applications in high power microwave amplifiers and possibly filtering structures.     

Alternative formulation for invariant optical fields: Mathieu beams

Based on the separability of the Helmholtz equation into elliptical cylindrical coordinates, we present another class of invariant optical fields that may have a highly localized distribution along one of the transverse directions and a sharply peaked quasi-periodic structure along the other. These fields are described by the radial and angular Mathieu functions. We identify the corresponding function in the McCutchen sphere that produces this kind of beam and propose an experimental setup for the realization of an invariant optical field.     

Modified null field method for P- and SV-wave scattering from a partially debonded fiber

The modified null field approach to elastic P- and SV-wave scattering (in plane strain) from a partially debonded fiber has been developed. The debonded region on the fiber surface is subjected to traction free boundary conditions, whereas the fiber is assumed to be in welded contact with the host medium elsewhere. Additional null field equations for an elliptical extension of the actual surface of the scatterer are introduced which require additional expansions for the field in the region between the extended mathematical boundary and the actual boundary of the scatterer. The numerical accuracy of the modified null field method has been tested for the case of a perfectly bonded fiber for P- and SV-wave incidence. Scattering cross-section plots are presented for different degrees of debonding.     

二元纯位相片实现半导体激光器到单模光纤的耦合

采用二元光学位相调制的基本思想，用二元纯位相型光学系统将椭圆高斯型激光束与单模光了佳耦合。采用两块纯 位相片来进行波前变换，将椭圆型高斯激光束转变成对称圆高斯激光束，并与单我纤的基模最佳匹配耦合。