Generalized switching properties of three-guide circular fiberarrays using coupled-mode analysis

A numerical method that utilizes coupled-mode theory to examine the switching properties of arrays formed by three fibers arranged in a circular configuration is developed. An error function that depends on the difference between the desired output conditions and those obtained for a given set of coupling and detuning coefficients is defined. To determine the values of these coefficients for a particular switching operation, the error function is minimized by varying the coupling and detuning coefficients. Results are presented for configurations that switch a signal from one fiber to another, equally split the power from a single input guide into two fibers at the output, and divide the power from a single input guide equally among all three output guides     

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.     

Computer Analysis of Gradually Tapered Waveguide with Arbitrary Cross Sections (Short Papers)

This short paper presents a general computer analysis of gradually tapered waveguide with arbitrarily shaped cross sectious. The technique combines coupled-mode theory with numerical methods for solving the uniform waveguide problem. The coupling coefficients are computed by using eigenvalues and eigenfunctions obtained numerically. The mode amplitudes are obtained either by numerical solution of a set of differential equations or from a closed-form solution. The applicability of this technique is illustrated by the analysis and measurement of two transitions. It is shown that theoretical prediction of coupled-mode amplitudes is reliable for gradual tapers where the flare angle is small. For large flare angles, more rigorous coupled-mode theory has to be employed.     

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     

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     

Coupling characteristics of eccentric arranged dielectric disk andring

An analysis is presented for a novel coupling configuration in which a circular dielectric disk and ring are arranged eccentrically. Whispering gallery (WG) mode coupling characteristics between the dielectric disk and ring are investigated. In this paper, a coupled-mode equation based on the Lorentz's reciprocity theorem is utilized. Distributed coupling coefficients and electric field distributions around the coupling region are obtained numerically through solving the coupled-mode equation. The theory described in this paper is confirmed by comparing measured electric field distributions with calculated ones. Electromagnetic powers flowed along the disk are also calculated. It is shown that coupling quantity of the eccentric configuration would be easily controlled by changing a radius of the disk or ring. The results obtained here will be used to design a WG mode coupled resonator for millimeter wave integrated circuits     

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     

Coupled waves on optical fibers by power conservation

Using only power conservation, the authors derive the coupling coefficients of coupled-mode theory and a new power conservation law. The derivation provides physical insight into the mechanism of wave coupling on optical fibers. It shows explicitly that the coupled-mode approximation is not generally applicable to Maxwell's equations, since the polarizability of perturbations is omitted. Examples include nonlinear fiber couplers and fiber nonuniformities, with emphasis on their polarization characteristics. An important class of unsolved problems is revealed     

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     

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.     

Cross power and crosstalk in waveguide couplers

The operation of two-guide couplers is re-examined to include the effects of cross power. Two two-guide coupler structures that operate as on-off switches are analyzed using both the exact eigenmodes of a slab model of the coupler and a coupled-mode theory that takes cross-power effects into account. The two coupler configurations analyzed are with the output guide on the same side as the input guide and on the side of the coupled guide. For the proper choice of the length of the coupler in the first case, the waveguide detuning can be adjusted so that ideal switching behavior can be achieved. For the second case, full on-off switching is not possible. Excellent agreement is found between results of the coupled-mode theory and exact slab analyses     

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     

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.     

Coupled modes of anisotropic polarization maintaining fibers

Wave coupling between the fundamental vector modal fields of two identical, polarization-maintaining, weakly guiding fibers is analyzed. An improved coupled-mode theory is used that also applies to anisotropic materials. Coupling coefficients that are the elements of a coupling matrix are calculated and presented as functions of the relative tilt angle between the two fiber principal axes. Exact analytical expressions for the eigenvalues and their associated eigenvectors are given in terms of the elements of this coupling matrix, for any relative tilt angle. Symmetry relations that are found between elements of the coupling matrix much simplify the presentation of the eigenvectors. A weak-coupling approximation of the eigenvector expressions, which is also given, retains their qualitative properties     

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.     

Mode coupling in circular-cylindrical system and its application tofingerprint resonators

Coupled-mode equations are given in the circular-cylindrical coordinate system for the evaluation of two-dimensional fingerprint resonators. The general form of coupled-mode equations is outlined. The coupling coefficients are found for transverse electric and transverse magnetic modes. The quality value Q is defined, and a numerical evaluation is given. A detailed derivation of the coupled-mode equations is given     

A comparative study of higher order Bragg gratings: coupled-mode theory versus mode expansion modeling

The modal reflectivity and loss of lamellar diffraction gratings to be used in distributed-feedback and distributed-Bragg reflector lasers were computed in dependence of wavelength, duty cycle and Bragg order. Different methods based on the approximate coupled-mode theory and and the exact bidirectional mode expansion modeling were compared and a good mutual agreement was found. The slab Green's function needed to compute the coupling coefficients can be approximated by that of a homogeneous unbounded medium with sufficient accuracy.     

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.<>     

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.     

Full Strain Tensor Treatment of Fiber Bragg Grating Sensors

Embedded fiber Bragg gratings can be subjected to arbitrary states of strain including shear strain. Such perturbations can cause coupling between polarization modes. Coupled-mode theory in Bragg gratings so far neglected this effect and only considered forward-backward coupling. Polarization mode coupling within a Bragg grating leads to interdependencies between Bragg reflection peaks which have so far been unaddressed. We formulate a full strain tensor treatment of fiber Bragg gratings, considering the coupling of the polarization modes within the grating. We give an approximation for the coupling coefficients affecting the polarization mode coupling and numerically solve the coupled-mode equations for representative states of strain. We show in which way shear strain affects the optical response of a grating and demonstrate how the fiber's beat length influences this characteristic.