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 formulation of multilayered and multiconductortransmission lines

A novel coupled-mode formulation for multilayered and multiconductor transmission lines is developed. In this formulation, the solutions to the original multiconductor system are approximated by a linear combination of eigenmode solutions associated with the isolated single conductor line located in an appropriate reference dielectric medium. The reciprocity theorem is used to derive the coupled-mode equations. The coupling coefficients are expressed in terms of the simple overlap integrals between the eigenmode fields and currents of the individual conductor lines. As a basic application, the dispersion characteristics of two identical coupled-microstrip lines are analyzed using the proposed coupled-mode theory. It is shown that the results are in very close agreement with those obtained by the direct Galerkin's moment method over a broad range of weak to strong coupling     

Analysis of optical fiber directional coupling based on theHE11 modes. I. The identical-core case

For pt.II see ibid., vol.8, no.6, p.832-7 (1990). The directional coupling between two identical single-mode optical fiber cores is analyzed by using the coupled-mode theory in the vectorial form based on the exact HE₁₁ modes. Analytical expressions for the coupling coefficients and the butt coupling coefficient appearing in the conventional and the recently formulated new coupled-mode equations are presented. The effect of the different polarizations on the coupling is considered. The accuracy of the conventional and new theories as applied to the two-core system is examined by comparing the coupling coefficients with exact numerical values. It is shown that as long as the exact HE₁₁ modes are used, the coupled-mode calculations behave well in predicting the coupling strength beyond the weakly guiding regime, and that the simpler conventional theory can provide satisfactory results when applied in the vectorial form     

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     

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.     

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 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.     

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     

Crosstalk and switching characteristics in directional couplers

The normal mode approach is used to calculate the fundamental limits on crosstalk and switching characteristics in directional coupler switches of the stepped delta beta type. The normal modes are calculated numerically as well as analytically using a method for computing the eigenmodes of two coupled, identical waveguides with an antisymmetric index perturbation. The results of the two methods are compared to one another as well as to the results of coupled-mode theory. It is shown that the theoretical minimum for the crosstalk level is very low for the stepped delta beta structure. Nonzero crosstalk was found in some cases where coupled-mode theory predicts zero crosstalk. This is attributed to the improper handling of discontinuities in coupled-mode theory     

Coupled waveguide antennas

A structure consisting of two coupled waveguides is proposed as a leaky-wave antenna. The analysis of this device is based on coupled-mode theory. Previous coupled-mode treatments are extended, since radiative loss is not neglected and the asymmetry of the structure is taken into account. Two cases are considered: the coupling parameter constant with distancezalong the antenna and the coupling as a function ofz. In both, the radiative attenuation is maintained constant. It is found that the first case permits two "normal modes" to propagate and, if both modes are excited, the patterns may be superposed to produce desirable overall results. An example is given of an approximation to a pencil beam pattern. By means of the second method, essentially any of the common excitation functions may be produced. An example is given of an antenna to achieve a cosine-on-a-pedestal excitation. Experimental results are given and compared to theory. Suggestions are presented for further work.     

Coupled multiple waveguide systems

The coupled-mode equations for systems of four and five planar coupled waveguides are investigated to determine transmission characteristics of interest. By proper selection of the coupling coefficients in a system of "synchronous" waveguides, full transfer can be achieved from one outermost guide to the guide on the opposite side. In the five-waveguide system, an excitation of the center guide can be fully transferred to a symmetric excitation of the outer guides. Sinusoidal modulation of the propagation constants leads to transfer functions with desirable properties for sampling applications. The analysis for synchronous waveguides is extended to waveguide systems of arbitrary order. Expressions for the coupling coefficients of anNth order system are given for each of the two types of transfer characteristics described above.     

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-Mode Analysis of Nonuniform Coupled Transmission Lines

The transmission-line equations describing propagation along coupled transmission lines are cast in coupled-mode form so that the roles of the different coupling coefficients and the impedance variation are more directly observable. Restrictions on the various parameters for obtaining directional coupling with nonuniform lines are then discussed and exact solutions for the coupling response of two nonuniform coupled lines with particular variations of the coupling coefficients are presented. The results obtained from the exact closed-form solutions should aid in the design of tapered couplers.     

Spectral characteristics of coupled-waveguide Fabry-Perotresonators and filters

Conventional Fabry-Perot resonators provide a narrow spectral width but lack the capability of longitudinal mode discrimination. A coupled-waveguide Fabry-Perot structure made of two parallel waveguides with reflecting mirrors at the ends is proposed for applications as mode selective resonator in single-mode diode lasers and as narrow-band wavelength filters. The interference of counter propagating waves from reflection by end mirrors and the coupling of waves between the two parallel waveguides contribute to the operation of this resonator structure. Thus, the device exhibits the attributes of both Fabry-Perot resonator and directional coupler. The coupled-mode theory of parallel waveguides is employed to analyze the structure. Both cases of identical and unidentical waveguides are examined. Resonance conditions and spectral characteristics are determined. It is shown that the coupled-waveguide Fabry-Perot resonator provides significant improvement in mode discrimination capability and longitudinal mode spacing over the conventional Fabry-Perot resonator     

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     

Characteristics of Coupled Microstrip Transmission Lines-I: Coupled-Mode Formulation of Inhomogeneous Lines

This paper consists of two parts. In Part I, coupled-mode theory is employed to determine the effects of reflection at the various ports and unequal inductive and capacitive coupling coefficients on the coupling and directivity of two coupled lines. Since couplers utilizing microstrip lines generally have unequal inductive and capacitive coupling coefficients, the results presented here should be useful in explaining the behavior of microstrip coupled lines. It is shown how the difference in the coupling coefficients leads to finite directivity and, under certain conditions, to "codirectional" instead of "contradirectional coupling." In Part II, the coupling coefficients and other parameters of various microstrip-line geometries are presented. Using these parameters in the results obtained here leads to an improved understanding of and design criteria for coupled microstrip lines.     

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     

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     

Analysis of optical fiber directional coupling based on theHE11 modes. II. The nonidentical-core case

For pt.I see ibid., vol.8, no.6, p.823-31 (1990). The analysis of the directional coupling between two single-mode optical fiber cores based on the exact HE₁₁ modes is extended to the cases involving nonidentical fibers. The coupled-mode theory in the vectorial form is used, and analytical expressions for the coupling coefficients and the butt coupling coefficient appearing in the conventional and the coupled-mode equations are presented. The accuracy of the conventional and new theories as applied to the two-core system is examined by comparing the coupled-mode predictions with the exact numerical values for different core-radius ratios, waveguiding strengths, polarization states, and core separations. It is shown that the errors in the coupled-mode theories increase as the two cores become more dissimilar. As long as the dissimilarity between the cores is kept small, the coupled-mode calculations using the HE₁₁ modes in predicting the coupling strength can be of satisfactory accuracy even when the individual guides are not weakly guiding ones. It is found that the new theory may give relatively large errors in the touching-core case with distinctly different core radii     

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