Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications – Asymptotic solutions

In two-dimensional optical waveguides with the varied refractive index, approximate analytic solutions of the leaky modes are derived by using an asymptotic analysis for both transverse electric (TE) and transverse magnetic (TM) modes. Numerical simulations show that the asymptotic solutions of the leaky modes are quite close to the exact ones. The results are useful in the eigenmode method, where the leaky modes appear if a perfectly matched layer (PML) is used to terminate the transverse directions of optical waveguides.     

Spectral representation of self-adjoint problems for layeredanisotropic waveguides

Layered waveguides with lossless anisotropic layers in the polar configuration are analyzed through the unifying concept of a real self-adjoint operator. For a suitable definition of two-vector transverse eigenfunctions, general properties such as orthogonality and completeness relations are derived. The linear operator formalism is applied to closed waveguides inhomogeneously filled with anisotropic materials, including crystals and gyrotropic media. As an extension of the former theory to open waveguides, a grounded uniaxial dielectric slab with a coplanar optic axis is also analyzed: as for open isotropic waveguides, a complete spectral representation including the surface (proper eigenfunctions) and the pseudosurface modes (improper eigenfunctions) is presented     

Network Representation and Transverse Resonance for Layered Anisotropic Dielectric Waveguides

First, the matrix wave impedance in an unbounded uniaxial lossless dielectric material is determined. Next, the transformation properties of the input impedance of a terminated anisotropic layer are established. It is then demonstrated that the boundary conditions in an anisotropic dielectric slab waveguide lead to a generalized transverse resonance condition involving the previously obtained matrix input impedances. Network equivalent representations are given for waveguides fabricated with dielectrics in polar and longitudinal orientations. The results show that a circuit approach to the analysis and design of planar anisotropic dielectric waveguides is feasible and practicable.     

Analysis of bilateral coplanar waveguides printed on anisotropicsubstrates for use in monolithic MICs

The spectral-domain method is applied to the analysis of bilateral (double-sided) coplanar waveguides that are printed on electric and/or magnetic anisotropic substrates. A nondecoupling approach is used to solve the coupled differential equations for the transverse propagation constants inside the substrate. The dyadic admittance Green's function is derived for both open and shielded bilateral coplanar waveguides, taking into account the anisotropy of the material. Finally, numerical results, describing the propagation characteristics of these structures, are presented for both electric and magnetic wall symmetries     

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     

Circular Waveguides Lined with Artificial Anisotropic Dielectrics

The propagation of electromagnetic waves inside circular waveguides lined with artificial anisotropic dielectric is investigated. Our investigation shows that the dominant hybrid electromagnetic (HEM/sub 11/ ) mode possesses a transverse deflecting field over the aperture of the structure and can be used as a transverse deflecting mode in a particle separator with ultrahigh energy. Expressions for power, attenuation, and transverse shunt impedance are obtained, and the effects of changing in loading on these various quantities are studied and presented in graphs.     

Transversely anisotropic curved optical fibers: variationalanalysis of a nonstandard eigenproblem

A novel variational functional is introduced for the analysis of curved open and closed waveguides. The theory is based on the variational principle for nonstandard eigenvalue problems. The present method is valid for the arbitrary waveguide cross section and arbitrary radius of curvature for closed waveguides; for open guides, the radius should be sufficiently large, because the method predicts the real part of the propagation constant, not the imaginary part, which gives the attenuation in curved open structures. The dielectric medium can be homogeneous or nonhomogeneous with transverse and/or longitudinal anisotropy. As an example of the method, curved isotropic and anisotropic single-mode fibers with two different kinds of anisotropy models are studied. The analysis includes field distributions, changes in the dispersion curves due to reformed geometry, and birefringence characteristics in curved anisotropic fibres     

The influence of light propagation direction on the stress-induced polarization dependence of silicon waveguides

This letter studies the effects of light propagation direction on the stress-induced polarization dependence of silicon-based waveguides. Silicon is an anisotropic material, so the stresses and the corresponding change of polarization dependence vary with the light propagation directions. The analyses show that when the light propagates in <010> directions on (100) silicon, the stress-induced changes of refractive index and the birefringence in effective index are about 20% more sensitive to the stresses than those when the light propagates in <011> directions.     

Orthogonality Relationships for Waveguides and Cavities with Inhomogeneous Anisotropic Media

A modfied reciprocity theorem forms the basis of development of orthogonality relationships for modes in waveguides and in cavities containing inhomogeneous, anisotropic media. In the lossless case certain of these may be interpreted in terms of power flow and energy storage. The special case of magnetized gyrotropic media is discussed for longitudinal and transverse magnetization.     

Efficient and accurate modeling of planar anisotropic microwavestructures by the method of lines

A new algorithm for the analysis of planar microwave structures with anisotropic substrates is proposed and substantiated. This algorithm is based on generalized transmission-line (GTL) equations, which are developed here for numerical algorithms. For the purpose of analysis, two different modal matrices for the discretized transverse electric and magnetic fields are calculated. Furthermore, impedance/admittance transformation formulas are developed with the help of the GTL equations for longitudinal sections and general junctions. Crossed discretization lines are used in the latter case. The materials are assumed to be biaxial or specific anisotropic. Special algorithms are developed for junctions consisting of more than two waveguides in the cross section and for bends. The proposed algorithm is verified by numerical results     

Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides

We describe a new full-vector finite difference discretization, based upon the transverse magnetic field components, for calculating the electromagnetic modes of optical waveguides with transverse, nondiagonal anisotropy. Unlike earlier finite difference approaches, our method allows for the material axes to be arbitrarily oriented, as long as one of the principal axes coincides with the direction of propagation. We demonstrate the capabilities of the method by computing the circularly-polarized modes of a magnetooptical waveguide and the modes of an off-axis poled anisotropic polymer waveguide.     

Performance of silicon micromachined waveguide at W-band

Fabrication and measurement of low-loss micromachined waveguides developed in (001) silicon through wet anisotropic etching is detailed. Such waveguides are candidates for use in submillimetre-wave systems owing to their compatibility with both efficient micromachined horn antenna elements and planar-based circuitry     

Dispersion characteristics of asymmetric coupled anisotropic dielectric waveguides using FDFD

The formulation is developed in the frequency domain and the finite difference method is used for the numerical solution of the scalar wave equation, written in terms of the transverse components of the magnetic field. As a result a conventional eigenvalue problem is obtained without the presence of spurious modes due to the implicit inclusion of the divergence of the magnetic field equal to zero. The formulation is developed to include biaxial anisotropic dielectrics with an index profile varying arbitrarily in the cross section of the waveguide under analysis. This formulation is then applied to the analysis of the influence on the dispersion characteristics of the dimensions of asymmetric coupled rectangular uniaxial anisotropic dielectric waveguides. As expected, the reduction of the height or the width of one of the rectangular dielectric waveguides causes the dispersion curves to move towards higher frequencies.     

Variational propagation constant expressions for lossyinhomogeneous anisotropic waveguides

Based on reciprocal relationships for the adjoint operator, we derive a variational formulation for the propagation constant satisfying the divergence-free condition in lossy inhomogeneous anisotropic waveguides whose media tensors have all nine components. In addition, with some advantages over previous representations, two variational formulations have been derived for waveguides with the transverse part of the media tensors decoupled from the longitudinal part. However, to obtain a variational formulation for a general lossy reciprocal problem the waveguide must be bi-directional. Each of the variational expressions results in a standard generalized eigenvalue equation with the propagation constant appearing explicitly as the desired eigenvalue. The stationarity of the formulations is shown. It is also shown that for a general lossy nonreciprocal problem the variational functional exists only if the original and adjoint waveguide are mutually bi-directional     

Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices

Dispersion characteristics of anisotropic guided-wave structures are analyzed by the edge-element method to eliminate spurious solutions. This approach can be applied to the cases in which the permittivity and permeability matrices are full. An eigenvalue equation which can provide direct solution for the phase constants is also derived when the tensors of the medium can be seperated into transverse and axial components. Numerical examples are presented for longitudinally magnetized, ferrite loaded waveguides and optical waveguides whose optic axis lies in xz- or yz- plane.     

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a Homogeneous Anisotropic Shell

A solution to the two-dimensional scattering properties of a conducting elliptic cylinder coated with a confocal homogeneous anisotropic elliptical shell is obtained. The transmitted field of the anisotropic shell is expressed as an integral equation based on waves with different wave numbers and different directions of propagation. The waves in all directions are represented as the eigenfunction expansion in elliptic coordinates in terms of Mathieu functions. In order to solve the nonorthogonality properties of Mathieu functions, Galerkin's method is applied and a matrix is required for the computation of unknown expansion coefficients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse electric (TE) polarization can be obtained in the same way. Some numerical results are presented in graphical forms. The result is in agreement with that available as expected when a coated elliptic cylinder degenerates to the coated circular one.      

Finite-Element Analysis of Waveguide Modes: A Novel Approach that Eliminates Spurious Modes

An efficient finite-element method for analyzing the propagation characteristics of a wide variety of waveguides is presented. A variational expression suited for the finite-element method is formulated in terms of the transverse electric and magnetic field components. In this approach all guided-mode solutions are real, while the spurious-mode solutions are not real. Therefore, discrimination of the spurious-mode solutions can be achieved merely by imposing the simple condition that guided-mode solutions be real. Three numerical examples, two for the isotropic case and the other for the magnetic anisotropic case, are carried out.     

Birefringence of passive multi-quantum-well semiconductor slabwaveguides

Effective refractive-indices of both TE and TM modes of passive semiconductor slab waveguides in the wavelength range near half the bandgap are measured and modeled. Comparisons among a bulk waveguide and several multiple-quantum-well (MQW) waveguides show that the layered structures of a MQW tend to enhance the birefringence. Also, for a waveguide there exists a birefringence maximum at a wavelength that depends on the guiding layer thickness. Theoretical calculations with a model of anisotropic guiding layers for MQW waveguides show good agreement with the experimental results     

Full-vectorial finite element beam propagation method withperfectly matched layers for anisotropic optical waveguides

Perfectly matched layer (PML) boundary conditions are incorporated into the full-vectorial beam propagation method (BPM) based on a finite element scheme for the three-dimensional (3-D) anisotropic optical waveguide analysis. In the present approach, edge elements based on linear-tangential and quadratic-normal vector basis functions are used for the transverse field components. To show the validity and usefulness of this approach, numerical examples are shown for Gaussian beam propagation in proton-exchanged LiNbO₃ optical waveguides. Numerical accuracy of the present PML boundary condition is investigated in detail by comparing the results with those of the conventional absorbing boundary condition (ABC)     

Generalized transmission line equations for bianisotropic materials

The generalized transmission line equations for bianisotropic materials are derived in this paper. These equations are obtained in a general orthogonal curvilinear reference system, taking properly into account the metric factors and decomposing the field vectors into two components: a transverse component and a longitudinal one, both with respect to a reference coordinate direction. Such a general formulation is very useful in order to analyze metallic/dielectric uniform bended waveguides. On the other hand, it is very appropriate when studying conformal integrated circuits by using the method of lines (MoL). Finally, the formulation here presented leads to a straightforward derivation of the transmission line equations in the spectral domain when studying conformal integrated antennas. As particular cases of the theory here presented, transmission line equations in orthogonal curvilinear coordinates are also derived for anisotropic, bi-isotropic, and isotropic materials.