About coupled-mode theories for dielectric waveguides

A critical examination is made of recent works on coupled-mode theory for dielectric waveguides with strongly overlapping fields. It is shown that there is no best formulation. In each case, explicit or implicit approximations lead to errors that are difficult to estimate. An investigation is made of the accuracy of coupled-mode formulas, in the case of strong guidance, for TM waves along coupled slabs or for HE11 modes along circular rods. Contrary to a previous prediction, there is no breakdown of coupled-mode formulas when the guidance is increased. The coupled-mode equations are applied to the problem of nonparallel waveguides in optical directional couplers. Comparison between coupled-mode predictions and beam-propagation-method simulations shows that bending effects in converging and diverging sections affect the accuracy. An improved coupled-mode theory is proposed in order to take these effects into account     

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     

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     

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     

Coupled-mode formulations for parallel-laser resonators withapplication to vertical-cavity semiconductor-laser arrays

Laser resonators, with laterally patterned end mirrors, are considered in the framework of coupled-mode theory. Two distinct coupled-mode formulations are presented. The first is applicable to cavities having high-reflectivity mirrors surrounded by a significantly lower reflectivity background. The second formulation applies to high-reflectivity mirrors with a slightly lower reflectivity background. Unlike coupling between parallel waveguides, the interaction between neighboring elements of the parallel cavities is achieved via diffraction, and the perturbation is of the boundary conditions rather than of the dielectric constant. These formulations are directly applicable to the case of arrays of coupled vertical-cavity semiconductor lasers     

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     

Application of Generalized Characteristic Vectors to Problems of Propagation in Clad Inhomogeneous Dielectric Waveguides

A transformation matrix that uses generalized characteristic vectors is used to convert Maxwell's equations into a set of loosely coupled equations for the wave amplitudes. This transformation is suitable for permittivity profiles with turning points. In earlier full-wave solutions to these equations, several special functions that account for the local features of the permittivity profile, especially near the turning points, were used to obtain appropriate expansions of the fields. The transverse field components, the propagation coefficients, as well as the phase and group velocities, are computed for both horizontally polarized (TE) and vertically polarized (TM) modes of the dielectric waveguides using the full-wave approach. These solutions are compared with analytic solutions for waveguides with special permittivity profiles. They are also compared with recently published results based on a perturbational approach.     

Improved coupled-mode equations for dielectric guides

An improved version of coupled-mode equations for parallel dielectric waveguides has been derived by using a newly found relationship that connects the propagation constants of the individual guides to the coupling coefficients via an overlap integral that measures the guides' proximity. The four parameters of these new coupled equations are simple functions of essentially one single quantity: the asynchronism of the individual guides properly normalized.     

Improved coupled-mode formulation based on composite modes forparallel grating-assisted co-directional couplers

An improved coupled-mode formulation based on the ideal modes of the coupled waveguides (ideal composite modes) is presented. In comparison with the formulation based on the ideal modes of the individual waveguides (ideal waveguide modes), the formulation in terms of composite modes is more rigorous and yields a more accurate grating period and coupling lengths. In addition, the radiation loss due to input and output junctions can in the composite-mode formulation. A new to the coupled-mode equations is derived in which all the spatial harmonics generated by the periodic grating are taken into account. The power exchange between the waveguides is examined by considering the input and the output conditions. The phase-matching conditions and the coupling lengths are calculated and compared with the analysis in terms of the waveguide modes. The grating period predicted by the waveguide-mode formulation agrees very well with that by the composite-mode formulation; however, dramatically different coupling lengths are predicted by the two formulations     

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.     

On the different formulations of the coupled-mode theory forparallel dielectric waveguides

Three different formulations of the coupled-mode theory for parallel dielectric waveguides are derived by substituting proper polarization vectors into a very general, unconjugated form of the reciprocity theorem. It is shown that the origins of errors of the three formulations can be clearly perceived through the derivation process. Power transfer characteristics predicted by the three coupled-mode theories are compared with the exact numerical results to show the relative accuracy of these theories     

A comprehensive study of discontinuities in chirowaveguides

We provide a comprehensive study of two- and three-dimensional discontinuities in chirowaveguides. The multimode coupled-mode method is an effective numerical approach to analyze this problem. After obtaining the coupled-mode equations, we diagonalize the coupling matrix to obtain a multimode scattering matrix rather than the usual two-mode approximation. We calculate the scattering properties of coaxial waveguides partially filled with lossy chiral media. Excellent agreement is observed between our results and those obtained by the mode-matching method. We also compare our results in the achiral case for dielectric material partially filled rectangular waveguide with experimental data and results obtained by the mode-matching method. Excellent agreement is again found. Based on our analysis, numerical and analytical results are displayed to provide physical insight into the problem. First, we discuss the effects of the chirality admittance on scattering properties and find that the sensitivity of the scattering parameters to chirality admittance increases as the chirality admittance increases. Second, we find the dielectric constant has a great influence on the scattering parameters. Third, we find the relative influence of height and width of chiral obstacles in rectangular waveguides.     

Improved coupled-mode theory of anisotropic optical waveguides

A coupled-mode equation for anisotropic waveguide systems of arbitrary cross section and general dielectric distribution is derived. Numerical results comparing the exact calculations to those of the method of Hardy et al. (Opt. Lett., vol.11, 742-4, 1986) show that the same accuracy can be obtained not only for TE, but also for TM mode coupling in the case of anisotropic waveguides, and the improved coupled-mode theory is applicable to the situation when moderately strong coupling occurs under the condition where the edge-to-edge separation of two coupled guides D₂ is about 0.1 μm     

Coupled mode theory of parallel waveguides

A new coupled mode formulation for parallel dielectric waveguides is described. The results apply to any guided modes (TE, TM, or hybrid) in waveguides of arbitrary cross-section, dissimilar index, and nonidentical shape. Additional index perturbations not included within the waveguides are encompassed by the theory. Propagation constants and mode patterns for the coupled modes computed according to this theory are shown to agree very well with numerical solutions for the system modes when the latter can be determined. Moreover, the new results are more accurate than those obtained from prior coupled mode formulations. It is shown that even for Iossless guides the coupling coefficients from waveguide"b"to"a"and from"a"to"b,"described by kaband kbarespectively, are not related by their complex conjugates if the guides are not identical.     

Scalar BPM analyses of TE and TM polarized fields in bent waveguides

We have applied the effective index method to reduce the two-dimensional (2-D) refractive index profile into the 1-D refractive index structure and modified the wave equations to obtain the paraxial wave equations. Then, transverse electric (TE) and transverse magnetic (TM) polarized fields in the curved single-mode planar waveguides are analyzed by using the scalar beam-propagation method (BPM) employing the finite-difference method with a slab structure. The bending loss in bent waveguides is analyzed for optical fields obtained from the BPM and comparisons are made between the loss for the waveguides with various radius of curvature and refractive index difference. The outward shift of the optical field, which is generated at the connection between a straight and a bent waveguide, is obtained from the results of calculation of location of the maximum optical intensity. The transition loss can be reduced by introducing an optimized inward offset at a straight-to-bend junction. The birefringence for TE and TM polarized fields in bent waveguides is calculated from the phase difference of the optical fields. The wavelength shift due to the birefringence of TE and TM polarized fields in bent waveguides is also calculated.     

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.     

Rigorous finite-difference analysis of coupled channel waveguideswith arbitrarily varying index profile

A rigorous finite-difference formulation for the hybrid-mode analysis of coupled diffused dielectric channel waveguides is presented. The method includes the two-dimensional continuous index profile variations directly in the finite-difference form of coupled equations and avoids the shortcomings inherent in the usual staircase approximations. The formulation in terms of the wave equation for the transverse components of the magnetic field leads to an eigenvalue problem where the nonphysical, spurious modes do not appear. The analysis includes the complete set of hybrid-modes, takes mode-conversion effects and complex waves into account, if they exist, and allows the calculation of dielectric channel waveguides with large index difference levels. Dispersion characteristic examples are calculated for coupled structures suitable for optical integrated circuits. The theory verified by comparison with results available from other methods     

Analysis of the electromagnetic scattering by thick gratings usinga combined FEM/MM solution

The problem of diffraction by a thick, conducting grating situated in an inhomogeneous dielectric slab is investigated using the generalized network formulation. This formulation combines the method of moments and the finite-element method, permitting the treatment of periodic elements of arbitrary cross section and inhomogeneous profiles. Solutions are presented for both transverse electric (TE) and transverse magnetic (TM) polarizations. Transmission gratings composed of rectangular conductors filled with dielectric materials of arbitrary profiles are studied     

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     

Coupled-zigzag-wave theory for guided waves in slab waveguidearrays

The guided wave in a slab waveguide array is treated as a set of coupled zigzag waves that propagate in the individual waveguides. This treatment leads to an exact dispersion relation for the TE and TM modes of the array, which can be expressed in a recurrence form and is easy to evaluate. Approximations for weakly coupled waveguides are discussed and compared with the coupled-mode theory. It is shown that the present theory can be cast in a form that closely resembles the coupled-mode theory