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     

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     

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     

A variational coupled-mode theory for periodic waveguides

A simple variational theorem for the dispersion relations and propagation constants of periodic waveguides is derived. The trial fields used in the variational formula incorporate all the Floquet components correct to first order. This yields a propagation constant in which errors enter only to fourth order in the trial field. The analysis yields a new coupled mode formulation which is shown to yield excellent agreement with exact analytic solutions (in l-D), and numerical simulations (in 2-D), for high-index contrast structures     

Improved coupled-mode theory for reciprocal anisotropic waveguides

An improved coupled-mode theory for a reciprocal anisotropic multiwaveguide system is derived. The general reciprocal anisotropic medium is described by a symmetric permittivity tensor that can have nonzero off-diagonal elements. The derivation is based on the generalized reciprocity relation. The coupled-mode equations are applicable to both lossy (gain) and lossless systems. For the special case of lossless systems, it is shown that the matrices in the coupled-mode equation are Hermitian, so that energy conservation is observed exactly. For the special case of a single anisotropic waveguide, the results also reduce to the previously derived solutions by D. Marcuse (1975). The improved coupled-mode theory in an anisotropic multiwaveguide system is illustrated with numerical examples     

An electromagnetic theory of dielectric waveguides with multipleembedded cylinders

We consider a dielectric waveguide or optical fiber having a real isotropic ambient refractive index. In addition, several isotropic and uniform embedded cylindrical regions of differing refractive index are considered. The radii, refractive indexes, and positions of the cylindrical subregions are arbitrary. We establish a rigorous method of determining the fields and propagation constants for the vector electromagnetic modes supported by this structure. We verify the accuracy of the method by comparing its results with those of other authors in particular cases, and exhibit direct demonstrations of convergence     

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.     

Relation between normal-mode and coupled-mode analyses of parallel waveguides

The analysis of parallel waveguides can be performed either by expressing the coupling of energy between individual, primary guides (coupled-mode approach) or by directly analyzing a structure consisting of all the guides and their surrounding media (normal-mode approach). A comparison of these two approaches, as well as the derivation of relations enabling the conversion of data, when possible, from one approach to the other, are presented in this paper. Finally, the results are used to investigate the dependence of the coupling coefficient on the spacing between the guides. Although the derivations are performed for 2-D (slab) configurations, the results pertain to 3-D guides as well, by applying an effective refractive-index method. It is also shown that normal-mode analysis predicts incomplete transfer of energy for strongly coupled guides, even for perfectly symmetric configurations.     

Vector coupled-mode equations for modulated periodic media

A simple method is described for deriving the vector coupled-mode equations for a dielectric medium which is characterized by a longitudinally varying permittivity composed of a number of closely spaced, discrete, wave numbers. The reflection coefficient for an amplitude modulated periodic slab is displayed.     

Variational coupled-mode theory of optical couplers

A scalar coupled-mode theory is developed from the variational principle. The formulation is variational in the sense that the mode parameters are adjustable and can be optimized subject to the coupling between the guides. The theory is applied to slab and fiber couplers. In comparison with the conventional scalar coupled-mode theory in which only the amplitudes of the modes in the individual guides are adjustable, the variational scalar coupled-mode theory predicts more accurate propagation constants and field patterns of the normal modes of the couplers     

Nonorthogonal coupled-mode theory of grating-assisted codirectionalcouplers

Grating-assisted codirectional couplers are analyzed by a nonorthogonal coupled-mode theory. The coupled-mode equations are solved exactly, and the effects of all the space harmonics generated by the periodic gratings are included in the analysis. More rigorous relations among the grating period, the grating height, and the coupling length are established. The power exchange between the guides is investigated as a function of the propagation distance and the wavelength. A comparison with previous approximate solutions is made and discussed     

Section-wise-exact coupled-mode theory of waveguidequasi-phase-matched second-harmonic generation

We derive exact mathematical expressions for both the amplitude and phase evolution of optical waves due to linear and nonlinear interactions in a waveguide device and propose a section-wise-exact coupled-mode theory analysis for quasi-phase-matched second-harmonic generation in the waveguide. Section-wise-exact coupled-mode theory is readily applicable to nonuniform quasi-phase-matched gratings and can be useful as a quantitative method in their analysis and design. Using the method, several numerical results are presented for second-harmonic generation in periodically poled LiNbO₃ channel waveguides. In addition, we introduce a split-step method which takes propagation losses into account and which can include linear and two-photon absorption and waveguide losses     

Graded-index planar dielectric waveguides

Properties of planar dielectric waveguides having a diffusion-induced index of refraction distribution are investigated. It is found that the spatial distribution of modes, group velocity, and mode spectrum of these guides differed qualitatively from the corresponding properties found in a slab waveguide. These results were experimentally verified by measuring the mode spectrum of a dielectric waveguide having an assumed complimentary error-function distribution of refractive index.     

An exact formulation of coupled-mode theory for coupled-cavitylasers

Coupled-mode rate equations for coupled-cavity lasers are derived using a novel approach. The method, based on the Mittag-Leffler theorem, is exact. The coupling coefficients are compared to those derived by several different approximations     

Analysis of rib dielectric waveguides

An analysis of rib dielectric waveguides based on a mode matching technique is presented. When the constituent slab guides support only one guided mode, it is shown that the cutoff condition for the higher order modes is the same as the result of the effective dielectric constant method. However, when the rib region is thick enough to support two guided slab modes, the cutoff conditions are significantly different. For this case universal design curves are produced and plotted. The results are also compared with another theory that exists to analyze such structures.     

Self-consistent vector coupled-mode theory for tapered opticalwaveguides

A nonorthogonal coupled-mode theory for nonparallel waveguides is derived. An orthogonal coupled-mode theory is developed based on the exact local array modes. The relation between the two formulations is established. Closed-form solutions are obtained for the two-guide synchronous tapered coupler. The formalism obeys power conservation     

Improved coupled-mode theory for nonlinear directional couplers

An improved coupled-mode equation for nonlinear directional couplers (NLDC) with Kerr-like nonlinear media is proposed. The method is based on the generalized reciprocity relation in which two sets of field solutions satisfying Maxwell's equations are for the NLDC and for the isolated nonlinear waveguide. That leads to a reasonable result that all the coefficients in the coupled-mode equations, including the coupling coefficient, become power dependent. For the NLDC with a self-focusing or self-defocusing nonlinear material, the authors examine how the coupling coefficient, the coupling length, and the guided power depend on the input power in the range of coupling stronger than in previous reports. It is found that the critical power at which the coupling length becomes infinity does not increase as much with the two guides for the case of self-focusing media     

Improved coupled-mode theory for anisotropic waveguide modulators

An improved coupled-mode theory for the propagation of modulated light waves in anisotropic dielectric waveguides is presented. Starting from Maxwell's equations, a partial differential equation is derived to describe mode-coupling between two normal modes which may propagate in anisotropic waveguide systems under modulation. The theory is applied to the analysis of typical waveguide modulators; examples for LiNbO₃ phase modulator, Mach-Zehnder, and directional coupler modulator are presented. The theory is applicable to both bulk and waveguide modulators/switches from DC to the high-frequency band. The only limitation is that the modulating wave has to propagate collinearly to the light waves     

Reformulation of the coupled-mode theory of multiwaveguide systems

We show that by proper reformulation the coupled-mode theory of Hardy and Streifer can be reconciled with that of Haus et al based on a variational principle. The new formulation has the merit of leading to simpler coupled-mode equations while giving marginally more accurate results.     

Step discontinuities on dielectric waveguides

Reflection, transmission and radiation properties of a step discontinuity on a dielectric surface waveguide are calculated. The method used involves bounding the open waveguide with perfect conductors. Results are compared with those given by previous authors.