Direct numerical integration of the wave equation is used to establish the validity of approximating the fundamental mode of graded-index fibers by a Gaussian function. We show that the fundamental modes of fibers, whose index profile can be expressed as a power law, are indeed very nearly Gaussian in shape (that is probably also true for graded-index fibers with convex profiles other than a power law). Graphs and empirical analytical expressions are presented for the optimum Gaussian beam width parameter and for the propagation constant of the fundamental mode.

Gaussian approximation of the fundamental modes of graded-index fibers

Propagating beam solutions for optical waveguides can be made to generate such mode-related properties as propagation constants, relative mode powers, and group delays with high precision and considerable flexibility. These quantities are needed in the analysis of optical fiber dispersion. The technique requires the generation of correlation functions from the numerical solutions of a wave equation. These correlation functions are in turn Fourier-transformed with respect to axial distance z. The resulting spectra display sharp resonances corresponding to mode groups, and the positions and heights of these resonances determine the previously mentioned mode properties. The spectral analysis is made highly accurate by the use of line-shape fitting techniques. With this method, mode group delays can be determined to a precision of +/-0.12 psec/km using a computation covering a 5-cm propagation path.

Computation of mode properties in optical fiber waveguides by a propagating beam method

An efficient and powerful technique has been developed to treat the problem of wave propagation along arbitrarily shaped single-mode dielectric waveguides with inhomogeneous index variations in the cross-sectional plane. This technique is based on a modified finite-element method. Illustrative examples were given for the following guides: (a) the triangular fiber guide; (b) the elliptical fiber guide; (c) the single material fiber guide; (d) the rectangular fiber guide; (e) the embossed integrated optics guide; (f) the diffused channel guide; (g) the optical stripline guide.

Single-mode optical waveguides.

Methods are developed for extracting from a numerical propagating-beam solution of a scalar wave equation the information necessary to compute the impulse-response function and the pulse dispersion for a multimode graded-index fiber. It is shown that the scalar Helmholtz equation and the parabolic wave equation have the same set of eigenfunctions in common and that the eigenvalues for the two equations are simply related. Thus one can work exclusively with the simpler parabolic equation. Both the mode eigenvalues (propagation constants) and mode weights, which are necessary for determining the impulse response, can be obtained with high accuracy from a numerical Fourier transform of the complex field-correlation function by the use of digital-filtering techniques. It is shown how a solution obtained in the absence of profile dispersion can be simply corrected for the presence of profile dispersion. In an illustrative example a graded-index fiber with a central dip in its profile is considered.

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Calculation of dispersion in graded-index multimode fibers by a propagating-beam method

The propagating beam method utilizes discrete Fourier transforms for generating configuration-space solutions to optical waveguide problems without reference to modes. The propagating beam method can also give a complete description of the field in terms of modes by a Fourier analysis with respect to axial distance of the computed fields. Earlier work dealt with the accurate determination of mode propagation constants and group delays. In this paper the method is extended to the computation of mode eigenfunctions. The method is efficient, allowing generation of a large number of eigenfunctions from a single propagation run. Computations for parabolic-index profiles show excellent agreement between analytic and numerically generated eigenfunctions.

Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method.

The propagation of electromagnetic surface waves along a radially inhomogeneous dielectric waveguide is investigated. The problem is formulated in terms of differential equations to be satisfied by the radially dependent parts of the electromagnetic field vectors. The dielectric waveguide is assumed to consist of a homogeneous cladding of infinite extent and a radially inhomogeneous core of higher permittivity. Numerical solutions of the differential equations in the core are obtained by two different methods, viz. by direct numerical integration and by substitution of an appropriate power series expansion. In the cladding the field is expressed in terms of modified Bessel functions. Imposing the boundary conditions at the interface of core and cladding, an equation for the unknown propagation coefficient is obtained. From this equation the propagation coefficients for the lower order modes are computed numerically. Numerical results are presented for some permittivity profiles of practical interest in single-mode transmission along optical fibres.

Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide

A full time-domain methodology is developed for analyzing the susceptibility of microwave electronic circuits to pulsed-field electromagnetic (EM) disturbances. The method is based on the consideration that micro-electronic circuits in their EM radiation behavior can be conceived as to consist of open (parallel-plate) resonators that are mutually coupled through their connecting feeds and are susceptible to disturbing incident fields through their apertures. In this respect, they are similar in behavior to the well-known ultra wideband patch antennas. The key issue to the method is to characterize each of these resonators by its natural modes of oscillation. Theoretically, these natural oscillations are adequately modelled through the time Laplace transform domain counterparts of the electromagnetic reciprocity theorem of the time-convolution type.

Pulsed-field EMI susceptibility analysis of microelectronic circuits - A full time-domain methodology

A consistent theory is developed by which the transmission properties of a uniform optical waveguide section of arbitrary cross section are characterized by the modal constituents as they are excited by equivalent surface-current distributions located in the end planes of the section. The representation leads in a natural way to orthogonality and completeness relations as well as to the appropriate normalization conditions.

H. Blok

Papers

Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide

Pulsed-field EMI susceptibility analysis of microelectronic circuits - A full time-domain methodology

Source-type representation for fields in optical waveguide sections and their consequences for the spectral field constituents