Variational approximations for higher-order modes of weakly-guiding fibres

Simple approximations for the fields and propagation constants of low-order modes on circular weakly guiding fibres are derived using variational techniques. The high accuracy of these approximations for quasi-parabolic profiles is demonstrated in the case of the Gaussian profile, for which there are no analytical solutions of the scalar wave equation.     

A perturbation approach to the propagation characteristics of inhomogeneous optical fibres with deformed core boundaries

The propagation characteristics of inhomogeneous optical fibres with different types of deformed core boundaries are analysed by a perturbative approach based on the scalar wave equation. The influence of the core deformation on the propagation constants and group delays is studied in detail for the truncated parabolic profile. We can conclude from this analysis that the difference in group delays due to deviation from circularity does not cause any difficult tolerance problems for realistic deformations and is generally negligible for practical optical fibres.     

Mode excitation in circularly symmetric graded-index optical fibers by focused gaussian beams

Field distributions and propagation constants of modes in weakly guiding circularly symmetric optical waveguides having arbitrary radial refractive index distribution are determinable from a variational analysis, which uses the solution of the scalar wave equation for the infinitely extended parabolic refractive index profile as a reference (Laguerre-Gaussian functions). The power coupling coefficients of the power, which is transferred from a focused Gaussian beam to an LP_vp-mode, depend on frequency and on four normalized launching parameters. Once the field-describing matrix equation has been solved numerically, closed-form expressions are obtained. The condition for optimal matching of the fundamental mode is given, and it is found that the maximum power excitation coefficient may be close to 100% even for substantially disturbed profiles.     

Mode excitation in circularly symmetric graded-index optical fibers by focused gaussian beams

Field distributions and propagation constants of modes in weakly guiding circularly symmetric optical waveguides having arbitrary radial refractive index distribution are determinable from a variational analysis, which uses the solution of the scalar wave equation for the infinitely extended parabolic refractive index profile as a reference (Laguerre-Gaussian functions). The power coupling coefficients of the power, which is transferred from a focused Gaussian beam to an LP_vp-mode, depend on frequency and on four normalized launching parameters. Once the field-describing matrix equation has been solved numerically, closed-form expressions are obtained. The condition for optimal matching of the fundamental mode is given, and it is found that the maximum power excitation coefficient may be close to 100% even for substantially disturbed profiles.     

Beam propagation in elliptical fibres

We apply an operator method, based on the scalar wave equation, to study the propagation of a Gaussian light beam in an optical fibre with an arbitrary, elliptically symmetric refractive-index profile. We discover that ray optical results, while not capable of predicting the rapid distortion of an optical beam in a non-ideal refractive-index profile, successfully simulate the general form of the optical power distributions, even in non-ideal profiles. We find that the propagation patterns for elliptical fibres with and without a strongly scattering central index dip show several unexpected similarities.     

渐变折射率波导传播特性微扰计算的代数递推公式

应用量子力学的超位力定理（HVT）和赫尔曼－费恩曼（Hellmann-Feynman）定理（HFT），导出了用微扰法计算渐变折射率平面波导传播常数各级近似值的代数递推公式，并给出了由传播常数近似值计算模场近似表达式的代数式。计算了几个典型实例，此法的特点是便于直接计算任意级近似的传播常数及模场分布，且可推广到二维波导的计算。     

非傍轴高斯光束传输方程解的探讨

由洛伦兹规范下光场满足的矢势和标势方程出发, 探讨了初始光场具有任意对称性的一般非傍轴光束在空间的传输情况。利用角频谱并结合傅里叶变换处理得到了非傍轴光束的传输方程。利用扰动方法并结合新近似取法得到了非傍轴横向光场的扰动形式解, 该解对于近场弱非傍轴情况下的效果比强非傍轴条件下要好。给出了光束非傍轴传输高阶(八阶)修正解。对于零阶情况, 非傍轴光束退化为傍轴光束, 传输方程也相应退化为傍轴传输的模型方程。高斯解为扰动参量的零阶修正解。     

Wave-matching method for mode analysis of dielectric waveguides

Frequently the cross-section of a longitudinally homogeneous dielectric waveguide may be decomposed into rectangles with constant permittivity. For points inside these rectangles the wave equation for modal fields is solved analytically by expanding into functions with harmonic or exponential dependence on the transverse coordinates. Minimization of a least-squares expression for the remaining misfit on the boundary lines allows us to determine propagation constants and fields for guided modes. Semivectorial calculations for two sets of rib waveguides and the centre sections of a directional coupler and an MMI device show very good agreement with results found in the literature. This revised version was published online in November 2006 with corrections to the Cover Date.     

A study of the propagation characteristics of W-type single-mode fibers using variational fields of single-clad single-mode fibers

A number of single and two parameter scalar variational fields approximating the fundamental mode of single-clad fibers are available in the literature. The present work aims at checking if these fields can also explain the propagation characteristics of W-type or double-clad single mode fibers with similar accuracies. To assess the accuracies, numerical solution of scalar wave equation is required. A method for solving the scalar wave equation numerically has also been proposed. Investigation shows that the single parameter fields are not very suitable for the W-type fibers. At least a two parameter variational field should be used.     

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。     

Propagation aspects of Mathieu–Gaussian beams in turbulence

Based on our recent source plane formulation, the propagation characteristics of Mathieu–Gaussian beams in turbulent atmosphere are investigated. In this connection, the average receiver plane intensity expression is deduced using the Huygens–Fresnel integral. Our results offered in the form of graphical illustrations reveal that, for some settings of source and propagation parameters, the center of the source beam is evacuated after propagation, while the initially smaller side lobes begin to grow. In a parallel development, the angular distribution of the beam also changes. At small Gaussian source sizes and transverse components of the wave vector, the source beam profile remains almost invariant throughout the propagation. The larger refractive index structure constant values cause the final Gaussian beam profile to be attained at earlier propagation distances. Smaller refractive index structure constants, on the other hand, do not change the beam profile substantially from that of free space.     

Scalar variational analysis of single mode fibers with Gaussian and smoothed-out profiles

A scalar variational analysis is presented to study the propagation characteristics of weakly guiding single mode fibers with gaussian and smoothed-out form of refractive index profiles. It is shown that the two parameter gaussian-exponential approximation for the fundamental mode describes the propagation characteristics with much higher accuracy as compared to the single parameter gaussian approximation.     

Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam

The problem of grating coupling of a focused incident beam under non-normal incidence into a slab waveguide is given a complete three-dimensional (3D) solution. The diffracted field is expressed as the Fourier integral of a regular part and of a singular part resulting from the existence of the coupled guided mode. A suitable expression of the field in the neighborhood of the pole and a rigorous definition of the modal field lead to a generalized coupled mode equation relating the incident field and the two-dimensional (2D) modal field propagating in the plane of the slab waveguide. The phenomenological parameters involved in the coupled wave equation: the propagation constant, the radiation coefficient as well as the modal field shape are derived from the exact treatment of plane wave diffraction in the same structure. The solution of the complete coupling problem is given in the particular case of a Gaussian incident beam, and of a high index step-index waveguide.     

On the effect of viscosity and thermal conductivity on sound propagation in ducts: A re-visit to the classical theory with extensions for higher order modes and presence of mean flow

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow.     

Electromagnetic waves in dielectrics with memory

We analyse electromagnetic wave propagation in a dielectric with memory of the Maxwell-Hopkinson type. We show that the components of the electric and magnetic fields satisfy two different scalar wave equations and we first look for their harmonic plane wave solutions. Then we prove that dielectrics with memory can also support approximate Courant-Hilbert waves. We discuss the equations to be solved to get all the components of the electromagnetic field from a scalar solution from each wave equation and TE, TM harmonic plane waves are explicitly given.     

单模及多模光纤折射率分布测量方法研究

通过对几种单模及多模光纤折射率分布测量方法的分析研究，得到单模光纤与多模光纤折射率分布测量方法的根本区别。由于单模光纤芯径比较小，因而只能用波动理论分析其传输机理，其中的远场法和近场法测量都是基于标量亥姆霍兹波动方程，即以单模光纤的基本传输理论进行测量；而多模光纤由于其芯径比较大，故而用射线理论分析其传输原理较为合理。多模光纤的折射近场法和近场扫描法均是以纤芯半径处数值孔径不同，对应的折射模和传导模不同为依据来进行测量的。     

Gaussian-beam propagation in generic anisotropic wave-number profiles

The propagation characteristics of a scalar Gaussian beam in a homogeneous anisotropic medium are considered. The medium is described by a generic wave-number profile wherein the field is formulated by a Gaussian plane-wave distribution and the propagation is obtained by saddle-point asymptotics to extract the Gaussian beam phenomenology in the anisotropic environment. The resultant field is parameterized in terms of e.g., the spatial evolution of the Gaussian beam's curvature, beam width, which are mapped to local geometrical properties of the generic wave-number profile.     

Photonic crystal fibers with material anisotropy

In this paper we are modeling the interplay of material and form birefringence in photonic crystal fibers. We introduce an efficient numerical method for the calculation of the modal structure. Our approach relies solving the fully vectorial wave equation for the transverse magnetic field and the respective propagation constants using a plane wave expansion. The method accounts for a simple form of material anisotropy. Our analysis is relevant to certain application areas, and in particular to fiber sensing, where material birefringence arises for instance due to transversally applied mechanical stress. We analyze the influence of material birefringence on the modal birefringence and the state of polarization of the fundamental mode.