Wave Scattering in Nonuniform Waveguides with Large Flare Angles and Near Cutoff

A set of coupled first-order differential equations for the wave amplitudes in nonuniform waveguides is developed. The coupling coefficients are regarded as differentiaI transmission and reflection scattering coefficients between two adjacent elementary radial waveguide sections. The analysis is an extension of an earlier quasi-optical solution. This set of coupled equations is compared with the familiar generalized telegraphist's equations which may be derived by considering the nonuniform waveguide to consist of elementary rectangular waveguide sections. The equations for the wave amplitudes derived in this paper are less coupled than the commonly used telegraphist's equations, and they may also be applied to waveguides with large flare angles and in regions at which the waveguide modes are at their respective cutoff cross sections.     

Scattering at a Junction of Two Waveguides with Different Surface Impedances

We consider junction of two cylindrical waveguides and derive the scattering matrix when a single mode is incident in one of the two waveguides. We are interested primarily in the case of two corrugated waveguides with different longitudinal impedances, but the analysis applies also to waveguides with nonzero transverse impedances. It is shown that, under certain general conditions the infinite set of equations specifying the junction scattering coefficients can be solved exactly by the residue-calculus method. Very simple expressions are then obtained between the scattering coefficients and the propagation constants gamma/sub n/ and y/sub i/ of the modes in the two waveguides. These expressions, obtained previously only in special cases, are direct consequences of certain simple relations derived here for the coupling coefficients between the modes of the two waveguides. In those cases in which the scattering coefficients cannot be determined exactly, we determine them approximately by a perturbation analysis.     

Contra-directional coupling in grating-assisted guided-wave devices

Contradirectional power coupling in grating-assisted guided-wave devices is studied by applying a vector nonorthogonal coupled-mode formulation. The coupled-mode equations are solved by a transfer matrix method. All the space-harmonics generated by the periodic grating are considered. The coupling can be understood in terms of the interference among the normal modes of coupled waveguides with a grating perturbation. Phase-matching grating periods for maximum reflections are equal to the beat lengths between the two normal modes involved in the coupling process. The reflections are built up constructively (Bragg reflection), resulting in stopbands in the spectral response. The expressions for the grating periods are obtained and compared with those derived from conventional phase-matching conditions     

A Perturbation Method for the Analysis of Wave Propagation in Inhomogeneous Dielectric Waveguides with Perturbed Media

This paper presents a perturbation method for determining the modes and the propagation constants of TE and TM waves in inhomogeneous dielectric waveguides whose index distributions depart from well-known profiles; e.g., a parabolic profile for which exact solutions can be obtained. Applying the variable-transformation technique to the wave equations, the wave-equation problem is transformed into the related-equation problem. The approximate solutions of the wave equations are obtained solving the related equation. The method is applied to the analysis of lower order mode propagation in a near-parabolic-index medium. The first-order field functions and the second-order propagation constants are given.     

A unified approach to coupled-mode phenomena

A unified approach is presented for the treatment of various coupled-mode phenomena in two parallel waveguides. This approach is summarized in a set of four coupled equations, which is derived directly from Maxwell's equations. The equations are further simplified when applied to special cases such as evanescent coupling and grating-assisted coupling between parallel waveguides [e.g., reduced to a set of two equations]. In particular, for evanescently coupled waveguides, the equations reduce to the familiar vectorial coupled-mode equations. For grating-assisted waveguides the equations agree with earlier treatments, although, in some cases, may include extra terms which were omitted previously. Considering the special case of perturbations in a single waveguide, the equations in the examples coincide with those given elsewhere in earlier works. The reduction to scalar equations or extension to multiwaveguide systems is straightforward     

Optimized offset to eliminate first-order mode excitation at thejunction of straight and curved multimode waveguides

Applications to equip semiconductor phased-array wavelength demultiplexers with nonbirefringent waveguides are increasing. In several applications, the waveguides are multimodal and elimination of first-order mode excitation is required for their design. The optimized offset for eliminating the first-order mode excitation at the junction of straight and curved waveguides with a two-dimensional (2-D) structure is analyzed theoretically. We assume that the waveguide has a laterally symmetric index profile. By the perturbation method, the offset is obtained to the first order in the inverse of the bending radius. The offset is simply expressed by the effective refractive indices of the straight waveguide     

Non-evanescent adiabatic directional coupler

A directional coupling mechanism based on an adiabatic coupling between three optical modes is suggested. The optical power transfer between two waveguides which are far apart is mediated by adiabatic coupling between zero-order optical modes of the individual waveguides and a high-order intermediate mode. The analytical model for an adiabatic three-mode coupling based on a scalar wave equation is presented. The directional coupling via the adiabatic mode coupling between copropagating modes is described and compared with a nonadiabatic directional coupling assisted by periodic perturbation. It is shown that adiabatic directional coupling has much less sensitivity to the mode parameters and to the wavelength     

Radiation loss of grating-assisted directional coupler

The coupling between the guided modes and between guided and radiation modes of two parallel slab waveguides forming a directional coupler in order is computed to determine the radiation losses introduced by the coupling grating. The problem is solved in two stages. First, the guided modes for each waveguide are computed separately and the radiation modes are only determined for the more complicated of the two waveguides, the one that is nearer to the grating. Modifications caused by the presence of the opposite waveguide are then taken into account by computing first-order correction terms. For a practical example of slab waveguides defined in GaInAsP, the authors find that the radiation losses per power exchange length remain below 0.02 dB for a rectangular grating depth of 0.01 μm     

Enhancement of power transfer in periodic array of optical waveguides via intermediate Bloch states

A directional coupling mechanism between different waveguides in a periodic array of waveguides is suggested. The optical power transfer between two different waveguides is mediated by the coupling between zero-order modes of two of the waveguides and the second band of the periodic structure. Analytical solutions for the no-detuning (narrow band) and far-from-resonance cases are presented. The far-from-resonance case is shown to resemble a simple two-mode system with complete optical power transfer between the two waveguides, coupled by localized gratings. The transfer is mediated by the second band of the periodic structure. The transition length depends strongly on the shape of the perturbation, and depends exponentially on the distance between the waveguides, yet it allows us to transfer power from one waveguide to another at such distances, for which the transition via conventional directional tunneling mechanism is impossible. Our analytical results are supported by numerical calculations carried out for a model problem with realistic parameters.     

Characteristics of coupling between parallel-plate waveguides with and without dielectric plugs

An analysis is presented of the coupling between parallel-plate waveguides excited in TE modes. Integro-differential equations are formulated for finite arrays of such waveguides with arbitrary widths and spacings. The waveguides, moreover, may be loaded with dielectric plugs having different dielectric constants and thicknesses. Solutions to these equations are effected by the method of moments. Extensive numerical data are obtained for the coupling between two waveguides, and their characteristics are examined in detail. The results show that in unloaded situations the coupling diminishes monotonically with increasings/lambda, wheresis the separation between the waveguides andlambdais the wavelength. At a given frequency, moreover, the coupling for largesdecreases asymptotically ass^{-3/2}. By contrast, an asymptotic dependence ofs^{-3/2}was uncovered earlier for TM-mode coupling. It is found that substantially different coupling behavior may result when the waveguides are loaded by dielectric plugs because of the excitation by the aperture discontinuity of higher order modes, which propagate inside the dielectric but are attenuated in the unloaded waveguide region. Of particular interest is the observation, under suitable conditions, of resonance characteristics in the coupling as functions of both the frequency and the thickness of the dielectric plug. These resonances are found to occur when the impedances of a certain higher order mode satisfy the transverse resonance condition, and thus are the manifestation of the resonances of such modes inside the cavity formed by the dielectric plug.     

正交矩形波导复合耦合槽特性分析

两个正交矩形波导公共宽边上偏离中心线的耦合斜槽称为复合耦合槽。考虑波导壁的厚度，根据等效原理和缝隙口面切向磁场的连续条件建立一对耦合积分方程，利用矩量法进行求解，给出复合耦合槽的谐振长度、散射参量随缝隙的倾角、偏离波导中心线距离的关系曲线     

Perturbation analysis of electromagnetic eigenmodes in toroidalwaveguides

The propagation of electromagnetic waves in a loss-free inhomogeneous hollow conducting waveguide with circular cross section and uniform plane curvature of the longitudinal axis is considered. The explicit solution of Maxwell's equations cannot be given in toroidal waveguides. For small curvature the field equations can, however, be solved by means of an analytical approximation method. In this approximation the curvature of the axis of the waveguide is considered as a disturbance of the straight circular cylinder, and the perturbed torus field is expanded in eigenfunctions of the unperturbed problem. Using the Rayleigh-Schrodinger perturbation theory eigenvalues and eigenfunctions containing first-order correction terms are derived for the full spectrum of all modes, including the degenerate ones. Complicated series expansions are obtained and are represented in closed form by means of the residue theorem     

Scattering loss of antiresonant reflecting optical waveguides

The scattering loss of two-dimensional antiresonant reflecting optical waveguides (ARROW) and of ARROW-B, which has a similar structure to ARROW and less polarization dependence, are analyzed by the first-order perturbation theory. Calculated results are compared with those of conventional three layer waveguides. Optimum design for the reduction of scattering loss of these ARROW-type waveguides is discussed. It was found that the scattering loss of ARROW-type waveguides is no larger than that of a conventional waveguide having a relative refractive-index difference, Δ of 2.5%, despite each interface of ARROW-type waveguides having a large Δ, normally larger than 20%. The optimum design for the reduction of essential radiation loss of ARROW is also optimum for the reduction of scattering loss     

A simple and accurate model analysis of strip-loaded opticalwaveguides with various index profiles

A simple first-order perturbation approach has been developed to study the propagation characteristics of strip-loaded diffused waveguides with various refractive index profiles. Propagation constants of the guided modes of rib waveguides and strip-loaded waveguides with exponential and Gaussian refractive index profiles are obtained. The results are in good agreement with those reported in the literature that were obtained by variational and numerical techniques. The presented technique provides analytical expressions for the modal field profile that should be useful in the design of various integrated optical devices     

Optimum Coupling for Random Guides with Frequency-Dependent Coupling

We obtain exactly the covariance of the signal-signal and signal-spurious mode transfer functions of the coupled line equations with two forward-traveling modes, white random coupling with statistically independent successive values (e.g., white Gaussian or Poisson coupling), and a coupling coefficient that varies with the frequency of the signals on the line. No perturbation or other approximations are made in this work. Time-domain statistics for the corresponding impulse responses are obtained for moderate fractional bandwidths. These results are extensions of a similar treatment for frequency-independent coupling coefficients, given in a companion paper. If the coupling were independent of frequency, the signal distortion would ultimately decrease as the coupling increased, approaching zero as the coupling approached infinity. The frequency dependence of the coupling coefficient prevents the distortion from approaching zero; the optimum coupling, which achieves minimum signal distortion, is independent of guide length. Millimeter waveguides and optical fibers with random straightness deviations have coupling coefficients inversely proportional to the frequency. The above results yield the optimum random straightness deviation for such a guide. More forward modes can be treated in a straightforward way by more complicated calculations.     

Coupling Between Dissimilar Waveguides

Investigators have used coupled-mode theory to analyze the coupling between identical waveguides; in such cases the coupling coefficients are found to be identical. If the waveguides differ, the coupling coefficients are asymmetrical and difficult to evaluate by strictly theoretical methods. An alternate approach to this case is considered in the present work. A pair of coupled-mode equations is first developed from a consideration of the permissible fields within the device. This clarifies the relationship between the coupled-mode theory and the more general classical electromagnetic theory by giving a careful definition of the coupled and the normal modes of a coupled structure. It is shown that the coupled-mode equations are an exact representation of the waveguide fields, although for engineering purposes it is often convenient to use approximate values of the coefficients of these equations. The mutual coupling coefficients are obtained from a two transmission-line model of the structure, with the actual coupling mechanism represented by a mutual impedance common to the two lines. For dissimilar lines, the ratio of the coupling coefficients is found to be equat to the ratio of the characteristic impedances. For the cases considered, this is the same as the ratio of the propagation constants of the uncoupled lines, which permits the coupling coefficients to be determined from relatively simple measurements. The adequacy of the theory has been confirmed by a series of experiments.     

Coupled-mode equations for pulse switching in parallel waveguides

The coupled-mode equations that describe the switching dynamics of optical pulses in two parallel waveguides are derived. These equations differ from the conventional ones in having extra coupling terms that arise from the dispersion properties of the coupling coefficient. It is shown that the coupling-coefficient dispersion can cause pulse distortion or even pulse breakup and in general produces much more significant effects than the group-velocity dispersion     

Coupling Between an Abruptly Terminated Optical Fiber and a Dielectric Planar Waveguide

The coupling between an optical fiber and a dielectric planar waveguide is analyzed when both guides are terminated abruptly and are facing each other. Mixed spectrum eigenwave representations of fields are employed inside the waveguides while Fourier integrals are utilized to describe the field in the space between the two guides. A coupled system of integral equations is derived by satisfying the boundary conditions on the terminal planes of both waveguides. A weak guidance approximation is assumed to facilitate the analysis. Numerical results are presented for several coupling geometries. Misalignment losses and coupling optimization phenomena are investigated.     

Analysis of nonlinear grating couplers by singular perturbationtechnique

The properties of the nonlinear output and input grating couplers are analyzed by using the singular perturbation technique with the multiple space scales. We first introduce the perturbation parameter concerned with the nonlinear parameter and the grating depth. After the wave functions are expanded, the perturbation solutions to satisfy the equivalent boundary conditions are derived. From the solvability condition to have nontrivial solutions on each perturbation order, the nonlinear equations to describe the power leakage of the guided wave due to the second-order coupling to the first-order waves are obtained. The dependence of the radiated field and the input efficiency on the power are discussed numerically     

Coupled-mode analysis of an asymmetric nonlinear directionalcoupler

A coupled-mode analysis of an asymmetric planar nonlinear directional coupler (NLDC) is presented by using the singular perturbation technique. The NLDC consists of a nonlinear waveguide with the core of Kerr-like medium and a linear waveguide situated parallel to each other. The effects of linear coupling and nonlinear modification of refractive index are treated to as small perturbations, and the modal fields of isolated linear waveguides are employed as the basis of propagation model. The self-consistent coupled-mode equations governing the power transfer are derived in analytically closed form. The representative numerical result for the input/output characteristics demonstrates that the asymmetric NLDC is useful for constructing a band-pass power-filter or a band-reject power-filter