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     

Low-loss optical branching waveguides consisting of anisotropicmaterials

Low-loss branching waveguides of the mode-conversion type consisting of anisotropic materials are proposed, and their basic wave-guiding characteristics are studied by means of coupled-mode theory. Two mode-conversion sections are introduced on both input and output sides of a conventional symmetric branching waveguide. Each arm of the branching waveguides is assumed to be a single-mode slab waveguide except for the tapered section. A coupled-mode system of equations describing mode-conversion phenomena with respect to the transverse magnetic (TM) mode in the branching waveguides is derived from the field expansion in terms of local normal modes. A Runge-Kutta-Gill method is used to numerically solve the coupled-mode equations. It is found that the proposed branching waveguides suffer mode-conversion losses to a much lesser extent than conventional branching waveguides     

Power exchange in tapered optical couplers

The power exchange between nonparallel optical waveguides or tapered couplers is analyzed by the coupled-mode theory based on local modes of the individual waveguides (local waveguide modes). A self-consistent nonorthogonal coupled-mode formulation is presented and transformed into an orthogonal form which is equivalent to the coupled-mode equations for the local modes of the coupled waveguides (local array modes). The coupled-mode equations are then solved numerically and the effect of the taper on the power exchange between the two guides is studied     

Coupled-mode analysis of two-parallel circular dielectricwaveguides using a singular perturbation technique

The evanescent coupling between two parallel circular dielectric waveguides is analyzed using a singular perturbation technique. The analysis is based on the vectorial wave formulation. The first-order coupled-mode equations are derived in an analytically closed form, which are rigorous in the sense that they satisfy the Maxwell equations and the boundary conditions for the composite waveguide system within the first-order perturbation. It is shown, in a general manner, that the two orthogonally polarized modes of the isolated waveguides yield the different coupling coefficients and the polarization effect is in proportion to the relative difference of permittivities of the core and cladding regions     

Assessment of coupled-mode theory for tapered optical coupler

The accuracy and the scope of validity of different coupled-mode formulations for a tapered optical waveguide coupler are assessed by comparison with the beam propagation method (BPM). It is demonstrated that the coupled-mode theory (CMT) based on local modes is valid only for small tilt angles, whereas an improved formulation that takes into account the wavefront-tilt effect yields highly accurate results for relatively large tilt angles. The radiation loss at the input and the output junctions can also be estimated by a simple mode-matching method combined with the improved CMT. The coherent coupling between two junctions,which is neglected in the coupled-mode formulations, is observed in the BPM simulations.<>     

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     

Improved coupled mode analysis of corrugated waveguides and lasers

Corrugated waveguides and lasers in resonant and non-resonant situations are analyzed by an improved coupled mode theory based on a set of the coupled mode equations for guided modes and radiation continuum. The distributed feedback (DFB) coefficient and the radiation loss coefficient are given in closed forms. The formulation can be applicable to arbitrarily shaped gratings and multilayer waveguide structures. The accuracy of the theory is examined by comparing it with Tamir's exact calculation for a nonresonant situation and also with Streifer's one for a DFB structure. Reasonable accuracy is obtained by the proper choice of the unperturbed waveguide parameter. The dependence of the two coefficients on the grating depth, the grating period, the guide layer thickness, and the refractive index difference between core and cladding layers is obtained for all Bragg orders up to the fourth, and for four typical grating shapes, namely, for rectangular, sinusoidal, symmetric triangular, and sawtooth gratings. Both the threshold gain of DFB lasers utilizing higher order Bragg reflection and the output coupling efficiency of grating beam couplers are also calculated for these parameters. A new multilayer structure for controlling the radiation loss is proposed and analyzed. This structure is suitable for the suppression of the radiation loss in DBR reflectors as well as for the improvement of the output coupling efficiency in grating beam couplers.     

A transfer matrix approach based on local normal modes for coupledwaveguides with periodic perturbations

A simple transfer matrix method is used to analyze power coupling and scattering in optical waveguide structures with periodic index perturbations. The transfer matrix is determined by a mode-matching technique for the local normal modes of the structures. This approach accounts for the change in the field patterns along the waveguide axis and is more rigorous than coupled-mode theory based on ideal modes. The distinct advantages of the method are that (1) it can be applied to structures with relatively strong index perturbations caused by large grating height and/or large index difference; (2) the TM modes whose transverse electric field is perpendicular to the index interface can be properly treated; and (3) the radiation loss due to scattering at the index discontinuities along the waveguide axis can be estimated. Analytical expressions for the power coupling in a grating-assisted codirectional coupler are derived, and numerical results are shown for some typical structures     

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     

Ray-optical determination of the coupling coefficients of gratingwaveguide by use of the rigorous coupled-wave theory

The ray-optics approach based on the rigorous coupled wave theory, called rigorous ray-optics method (RROM), is developed for the calculation of backward coupling coefficients of grating waveguide devices. The coupling coefficients of several grating structures, such as rectangular, sinusoidal, triangular, and trapezoidal shapes, are evaluated by the RROM, and they are compared with those obtained by two conventional methods of the ray-optics method (ROM) and the coupled-mode method (CMM). In the case of rectangular gratings, the coupling coefficients are evaluated in more detail by varying grating depth and duty-cycle. We have found that the RROM gives us more exact solutions for the backward coupling coefficients of even arbitrary shapes of diffractive grating waveguides than the other two conventional methods     

A new approach to grating-assisted couplers

A novel approach to the analysis and design of grating-assisted directional couplers is proposed. Power exchange between the waveguides is maximized through phase matching of two power-orthogonal modes of the parallel coupler. It is shown that either complete power transfer or zero crosstalk can be achieved at two different coupling lengths even when the two wavelengths are strongly coupled and/or close to synchronism. An analytical solution to the coupled-mode equations is obtained for the grating of rectangular shape. A grating-assisted coupler made of two slab waveguides is examined as an example to illustrate the salient features of the scheme     

On phase matching and power coupling in grating-assisted couplers

Two formulas of phase-matching condition for grating-assisted couplers in the literature are compared. The power exchange between the waveguides is examined by a numerical solution to the coupled-mode equations and a direct simulation by the beam propagation method. It is demonstrated that the phase-matching of the normal modes of the coupler (array modes) maximizes the power coupling. The approximate phase-matching condition for the modes of the individual waveguides (waveguide modes), however, leads to only partial power exchange     

Hyperbolic Secant Coupling In Overmoded Waveguide

This work presents a new solution of the coupled-mode equations for a hyperbolic secant spatial variation of the coupling between two modes. An analytic expression is given for the transmission coefficient for arbitrary complex differential propagation constant and coupling strength. The expression is particular simple in the case when the differential attenuation between the modes is negligible. Design curves are presented in terms of normalized parameters. The hyperbolic secant coupling may be truncated and still yield virtually the same transmission as for infinite couping length. The required coupling length is indicated by a comparison of the ideal expression with the results of numerical integration of the coupled-mode equations. Hyperbolic secant coupling can be particularly useful for the design of short low-loss broad-band bends, twists, and mode-selective couplers in overmoded waveguide. Results of tests on 90-degree bends in rectangular and corrugated circular waveguide are consistent with the theory.     

Coupled-mode equations for dielectric waveguides based on projection and partition modal amplitudes

There are two objectives in this paper. First, it is pointed out that the coupled-mode theory for parallel dielectric waveguides can have two different formulations, one based on the projection modal amplitudes and the other based on the partition modal amplitudes. The theory is then derived by considering the projection modal amplitudes of the total coupled-system field and including the integrated overlap of the individual waveguide modes. Second, the existing two formulations of the conventional coupled-mode theory are shown to be consistent with each other, following naturally from the discussions in the first part. Coupled-mode equations including the mode overlap integrals are written in terms of the projection modal amplitudes. They are appropriate to power calculations and are equivalent to those equations recently derived by other authors. It is emphasized that using equivalent initial conditions is essential for the different formulations to give identical predictions of power coupling.     

锥形光波导耦合特性研究

基于耦合模理论,数值仿真了两种耦合结构(FW和WF)下的耦合损耗随锥形光波导芯层尺寸的变换规律;进一步分析了两种耦合结构下锥形光波导端面粗糙度对其模式耦合损耗特性的影响,表明不同模式其端面散射损耗不同;实验结果表明,锥形光波导的输入端芯层尺寸、输出端芯层尺寸及光波导长度,对FW结构下的耦合损耗影响较小,对WF结构下的耦合损耗影响较大。     

Wavefront-tilt effect in nonparallel optical waveguides

The wavefront-tilt effect in nonparallel optical waveguides is studied theoretically. A self-consistent coupled-mode formalism based on guided normal modes of the individual waveguides is developed, and the wavefront-tilt effect due to the nonparallel structures is accounted for. The wavefront-tilt caused by junctions at input and output is considered by using a mode-matching method. The power exchanged between two slab waveguides separating at an angle is investigated by an analytical solution to the coupled-mode equations     

Concept of multiorder multimode resonant optical filters

The idea of implementing optical filters by coupling evanescent waves from several diffracted orders into multiple leaky waveguide modes is studied theoretically. Using a dielectric thin-film structure consisting of a coupling grating placed between adjacent waveguides, guided-mode resonance filters exhibiting multiple reflection peaks within a specified wavelength range can be obtained. These peaks originate in the resonant waveguide modes that are excited by the diffracted waves dispersed by the grating. It is shown that this device can be used to realize multiwavelength as well as wide-band spectral filters.     

A variational coupled-mode theory including radiation loss forgrating-assisted couplers

We calculate higher order effects such as frequency shifts and radiation loss in grating waveguides and grating couplers. The approach follows a vector variational principle which incorporates the Floquet components correct to first order. It is unnecessary to bring radiation modes into the formalism. This yields a simple coupled-mode formulation which includes attenuation due to radiation loss, and is correct to third order in the index contrast of the grating. Numerous examples are given which show excellent agreement with rigorous numerical simulations     

Optical waveguide directional coupler measurements using amicrocomputer-assisted TV camera system

Characterization of optical waveguide directional couplers using a microcomputer-assisted TV camera system is reported. A fluorescent technique is employed to visualize wave propagation in the optical waveguides, which exhibit small scattering. The method is simple and reproducible, and can be used to estimate waveguide losses and evaluate various optical devices such as waveguide lenses and waveguide interferometers on substrates. As an example of its application, parallel and nonparallel dual-chamber directional couplers and parallel triple-channel directional couplers are measured. Power transfer between the coupled channel waveguides is observed clearly, and coupling coefficients of the couplers are obtained nondestructively. The measured performance of the couplers is in good agreement with that predicted by coupled-mode analysis     

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.