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     

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     

Radiation method for the measurement of mode-conversion levels

A novel method for the determination of mode-conversion levels in overmoded waveguides is described. It relies on a swept-frequency measurement of the perturbation of the dominant-mode radiation pattern of the open-ended waveguide, owing to the presence of higher-order modes. Results are reported for mode-conversion effects in corrugated waveguides at frequencies near 9 GHz. Excellent agreement with the trapped-resonance method is demonstrated, although the present technique has advantages over this method when the mode excitation level is high or the waveguide length is imprecisely known.     

Full-vectorial wave propagation in semiconductor optical bendingwaveguides and equivalent straight waveguide approximations

Device characteristics of optical polarization rotators are founded upon the vector properties of the Maxwell equations. Recently, a bending waveguide based polarization rotator has been proposed and demonstrated. To provide a rigorous basis for the analysis and design of this polarization rotator, the full-vectorial wave equations for both E&oarr; and H&oarr;-field in bending waveguides are derived. It is found from these wave equations that under a broad range of circumstances, a bending waveguide can be analyzed using the equivalent straight waveguide approximation. Details of the model for optical polarization rotators, which is based on the coupled-mode theory, will be described in a companion paper     

Modeling and design of bending waveguide based semiconductor polarization rotators

A theoretical model for bending waveguide based semiconductor polarization rotators has been established, which is based on the full-vectorial wave equations for bending waveguides and coupled-mode theory. Calculation results obtained using this model and measurement data are found to be in good agreement.     

Analysis of Optical Waveguide Consisting of a Square-Law Lenslike Medium and its Analogies to Circular TE/sub 01/ waveguide

Propagation behavior of light beams along sinusoidal and serpentine bends as well as circular bends and linearly tapered bends of optical waveguides consisting of a square-law Ienslike medium is investigated in detail, both theoretically and numerically, on the basis of the approximate wave theory. A new design method of the circular bend for removing the effects of the bend is proposed and numerical results are presented. The divergence phenomena of the beam trajectory in both the sinusoidal and serpentine bends of the optical waveguide are discussed in comparison with mode-conversion phenomena occurring in the circular TE/sub 01/ waveguide with the same bends. Several design conditions to eliminate undulations of the beam trajectory and/or the spot size which would occur at a circnlar bend of the optical waveguide are also studied, and interesting analogies to the design conditions proposed so far to prevent mode-conversion losses at a circular bend of the TE/sub 01/ waveguide are shown.     

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 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     

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 Analysis of an NRD-Guide Coupler Based on Singular Perturbation Technique

A coupled-mode formulation for an NRD-guide coupler is presented using the singular perturbation technique. The first-order and second-order perturbations are taken into account in the analysis and the coupled-mode equations based on the eigenmodes of each waveguide in isolation are derived. The propagation constants obtained by these equations are compared with those by the exact theory, conventional coupled-mode theory, and improved coupled-mode theory. The numerical results of present formulation are in good agreement with the exact theory and superior to those of the other formulations.     

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     

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     

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     

A unified approach for the coupled-mode analysis of nonlinearoptical couplers

A unified approach for the coupled-mode analysis of nonlinear optical couplers is proposed. This approach is basically an extension of the work of [Haus, vol. 5, p. 16, 1987] to include optical nonlinearity. After the nonlinear coupled-mode theory is established, various basis functions are used as trial fields to derive coupled-mode equations. It is found that two published coupled-mode theories can in fact be deduced from the proposed one with individual linear or nonlinear waveguide modes serving as trial fields, thus making the coupled-mode equations from variational principle and reciprocity theorem equivalent. Coupled-mode equations based on system modes are also presented. Furthermore, analytical and/or numerical methods for solving coupled-mode equations are included. More research and discussion can be conducted based on the knowledge addressed in this paper     

Vector coupled-mode theory of dielectric waveguides

A consistent derivation of a system of vector coupled-mode (VCM) equations for parallel dielectric waveguides is presented and compared with earlier versions of the improved coupled-mode theory (ICMT). As a validity test, it is shown that the effectively scalar transverse electric and transverse magnetic (TM) coupled-mode (CM) equations are direct limits of our full VCM formulation. In particular, our formulation does not lead to the fundamental error found with earlier coupled-mode theories (CMTs) in a case of TM fields. Functional equations of our VCMT are consistent with Maxwell's equations and lead to higher precision. They can be applied to complicated arrays of strongly coupled parallel dielectric waveguides with true vectorial behavior.     

Analysis of the coupling characteristics of a tapered coupledwaveguide system

Optical coupling characteristics between one uniform waveguide and one tapered waveguide are discussed. This coupling structure is analyzed by utilizing coupled-mode theory and system step approximation. Based on numerical investigation, various coupling characteristics are presented. It is shown that the proposed structure can be used as a waveguide-type optical power divider by selecting an appropriate taper slope for the tapered waveguide or by separating the coupled waveguides properly for a desired power-dividing ratio     

Power Transfer in a Large Parallel Array of Coupled Dielectric Waveguides

This paper presents a compact analytical solution to the conventional coupled-mode equations for a large parallel coupled array of identical dielectric waveguides. Such arrays could be used as couplers in millimeter-wave or fiber-optic communication. The approach here is to characterize the array response to a signal injected into single waveguide first. This approach drastically simplifies the double Fourier sums called for in dealing with multiple waveguide injections. Regardless of the size of an array, the response consists of a series of Bessel functions, six of which suffice to give good accuracy. The general solution is then obtained by a linear superposition of responses when multiple waveguides are injected. Quasi-3-D profiles of waves propagating and spreading in a large array of 100 coupled guides can be quickly displayed using Microsoft Excel. Remarkably clear characteristics of power transfer such as self-focusing and boundary reflection are observed. Since the solution involves Bessel functions of high orders that diminish rapidly in value, the solution for very large arrays can be computed just as efficiently as for smaller arrays.     

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     

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     

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.