Single-mode optical waveguides and directional couplers with rectangular cross section: a simple and accurate method of analysis
TL;DR: In this article, a new method for obtaining the propagation characteristics of single-mode optical waveguides with rectangular cross section is presented, which is based on the scalar variational principle using the cosineexponential trial field.
Abstract: A new method for obtaining the propagation characteristics of single-mode optical waveguides with rectangular cross section is presented. The method is based on the scalar variational principle using the cosine-exponential trial field. This form of trial field leads to the definition of equivalent guiding structures consisting of homogeneous slab waveguides, which are then used to obtain, in a very simple way, the vector modes of a rectangular waveguide and the coupling characteristics of directional couplers consisting of two rectangular waveguides. The authors have also included the results of some representative calculations along with the results obtained using other approximate methods and the exact numerical method. A comparison shows that the method is much more accurate than the existing approximate methods, particularly in the region of single-mode operation. >
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TL;DR: Numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments are reviewed, ranging from the specialized ones for analysing restricted classes of waveguide to the most general ones for analyseing inhomogeneous, arbitrarily-shaped, anisotropic waveguiding.
Abstract: This paper reviews numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments. Six groups of methods are reviewed, covering (1) the finite-element method, (2) the finite-difference method, (3) the integral-equation method, (4) methods based on series expansion, (5) approximate methods based on separation of variables, and (6) methods that do not fit into the above groups, ranging from the specialized ones for analysing restricted classes of waveguides to the most general ones for analysing inhomogeneous, arbitrarily-shaped, anisotropic waveguides. Some suggestions with regard to the selection of methods for particular applications are given.
178 citations
TL;DR: In this article, a domain integral equation approach to computing both the propagation constants and the corresponding electromagnetic field distributions of guided waves in an integrated optical waveguide is discussed, and numerical results for various channel and ridge waveguides are compared with those of other methods where possible.
Abstract: A domain integral equation approach to computing both the propagation constants and the corresponding electromagnetic field distributions of guided waves in an integrated optical waveguide is discussed. The waveguide is embedded in a stratified medium. The refractive index of the waveguide may be graded, but the refractive indices of the layers of the stratified medium are assumed to be piecewise homogeneous. The waveguide is regarded as a perturbation of its embedding, so the electric field strength can be expressed in terms of domain integral representation. The kernel of this integral consists of a dyadic Green's function, which is constructed using an operator approach. By investigating the electric field strength within the waveguide, it is possible to derive an integral equation that represents an eigenvalue problem that is solved numerically by applying the method of moments. The application of the domain integral equation approach in combination with a numerically stable evaluation of the Green's kernel functions provides a new and valuable tool for the characterization of integrated optical waveguides embedded in stratified media. Numerical results for various channel and ridge waveguides are presented and are compared with those of other methods where possible. >
41 citations
TL;DR: In this paper, the cross-section of a longitudinally homogeneous dielectric waveguide is decomposed into rectangles with constant permittivity, and the wave equation for modal fields is solved analytically by expanding into functions with harmonic or exponential dependence on the transverse coordinates.
Abstract: Frequently the cross-section of a longitudinally homogeneous dielectric waveguide may be decomposed into rectangles with constant permittivity. For points inside these rectangles the wave equation for modal fields is solved analytically by expanding into functions with harmonic or exponential dependence on the transverse coordinates. Minimization of a least-squares expression for the remaining misfit on the boundary lines allows us to determine propagation constants and fields for guided modes. Semivectorial calculations for two sets of rib waveguides and the centre sections of a directional coupler and an MMI device show very good agreement with results found in the literature.
40 citations
TL;DR: In this article, the authors discussed the approximate methods used for obtaining scalar guided modes of optical waveguides, including the perturbation method, the variational method including the Rayleigh-Ritz method, and the Galerkin and the collocation method.
37 citations
TL;DR: In this paper, some trial fields are proposed for obtaining the propagation characteristics of single-mode inhomogenous planar waveguides, based on the scalar variational principle.
Abstract: Some trial fields are proposed for obtaining the propagation characteristics of single-mode inhomogenous planar waveguides, based on the scalar variational principle. A comparison with other trial fields and the exact numerical results shows that the authors' results are more accurate than those obtained by the existing trial fields. The proposed trial fields also agree favorably with the real inhomogeneous index profiles. >
22 citations
References
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TL;DR: In this paper, the authors studied the transmission properties of a guide consisting of a dielectric rod with rectangular cross section, surrounded by several dielectrics of smaller refractive indices.
Abstract: We study the transmission properties of a guide consisting of a dielectric rod with rectangular cross section, surrounded by several dielectrics of smaller refractive indices. This guide is suitable for integrated optical circuitry because of its size, single-mode operation, mechanical stability, simplicity, and precise construction. After making some simplifying assumptions, we solve Maxwell's equations in closed form and find, that, because of total internal reflection, the guide supports two types of hybrid modes which are essentially of the TEM kind polarized at right angles. Their attenuations are comparable to that of a plane wave traveling in the material of which the rod is made. If the refractive indexes are chosen properly, the guide can support only the fundamental modes of each family with any aspect ratio of the guide cross section. By adding thin lossy layers, the guide presents higher loss to one of those modes. As an alternative, the guide can be made to support only one of the modes if part of the surrounding dielectrics is made a low impedance medium. Finally, we determine the coupling between parallel guiding rods of slightly different sizes and dielectrics; at wavelengths around one micron, 3-dB directional couplers, a few hundred microns long, can be achieved with separations of the guides about the same as their widths (a few microns).
1,620 citations
TL;DR: In this article, a computer analysis of the propagating modes of a rectangular dielectric waveguide is presented, based on an expansion of the electromagnetic field in terms of a series of circular harmonics multiplied by trigonometric functions.
Abstract: This paper describes a computer analysis of the propagating modes of a rectangular dielectric waveguide. The analysis is based on an expansion of the electromagnetic field in terms of a series of circular harmonics, that is, Bessel and modified Bessel functions multiplied by trigonometric functions. The electric and magnetic fields inside the waveguide core are matched to those outside the core at appropriate points on the boundary to yield equations which are then solved on a computer for the propagation constants and field configurations of the various modes. The paper presents the results of the computations in the form of curves of the propagation constants and as computer generated mode patterns. The propagation curves are presented in a form which makes them refractive-index independent as long as the difference of the index of the core and the surrounding medium is small, the case which applies to integrated optics. In addition to those for small index difference, it also gives results for larger index differences such as might be encountered for microwave applications.
549 citations
"Single-mode optical waveguides and ..." refers background or methods in this paper
...5, for which the exact results are available [1]....
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...Of these the circular harmonic analysis by Goell [1] requires extensive numerical computations, and the closed form solution given by Marcatili [9] essentially neglects the modal field in the corner regions, thereby resulting in poor accuracy....
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...The numerical techniques, such as circular harmonic analysis [1], direct numerical integration of the field equations [2]-[3], finite element anlaysis [4], field expansion in orthogonal functions [5]—[8], etc....
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...2 along with the results obtained by using other approximate methods and the exact numerical results [1]....
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...3, along with the results obtained by using Marcatili's method [9] and the exact numerical results of Goell [1]....
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01 Jan 1975
TL;DR: In this article, the theory of dielectric waveguides is discussed and a discussion of the fabrication of planar waveguide structures is presented, even though most of the fundamentals are applicable to all waveguide types.
Abstract: Dielectric waveguides are the structures that are used to confine and guide the light in the guided-wave devices and circuits of integrated optics. This chapter is devoted to the theory of these waveguides. Other chapters of this book discuss their fabrication by such techniques as sputtering, diffusion, ion implantation or epitaxial growth. A well-known dielectric waveguide is, of course, the optical fiber which usually has a circular cross-section. In contrast, the guides of interest to integrated optics are usually planar structures such as planar films or strips. Our discussion will focus on these planar guides even though most of the fundamentals are applicable to all dielectric waveguide types.
464 citations
TL;DR: The effective index method for calculating waveguide mode dispersion is reviewed and applied to uniform rectangular optical waveguides with both small and large index differences and is shown to be at least as accurate as other approximate techniques.
Abstract: The effective index method for calculating waveguide mode dispersion is reviewed and applied to uniform rectangular optical waveguides with both small and large index differences. The results are shown to be at least as accurate as other approximate techniques. The effective index method is then applied to channel waveguides assuming 1-D and 2-D diffusion. Channel waveguides without sideways diffusion are shown to be described by the method using a normalized notation and previously published universal dispersion curves. Two-dimensional diffusion theory is applied to treat the case of isotropic sideways diffusion. A new, normalized, 1-D universal chart is obtained which in conjunction with previous results defines waveguide mode dispersion in isotropically diffused 2-D channels.
457 citations