Charles Vassallo

Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide

The reflectivity of a multilayer dielectric coating deposited upon the end facet of a symmetrical dielectric slab waveguide is investigated for both TE and TM waves. The integral equation satisfied by the field in the junction plane is solved numerically through a Fourier representation and an iteration process. The process converges rapidly and can easily provide a solution of as many as five or six exact digits in the case of problems involving semiconductor optical waveguides. In this case of sufficiently weak guidance, the integral equation can be simplified considerably, leading to an explicit solution for the Fourier transform of the field. This approximation greatly simplifies the computation of the reflectivity but is accurate enough to meet the current needs for the design of antireflection coatings.

Interest of improved three-point formulas for finite-difference modeling of optical devices

The accuracy of finite-difference computations is limited by the errors inherent in the classical three-point formula for ∂x2 (of order h2) and in the discretization of discontinuities in the refractive index (of order from h0 to h2). Various improvements are reviewed and analyzed, both for normal-mode derivations and for the beam propagation method (BPM). For the BPM it seems that sophisticated discretizations for discontinuities have a limited interest, whereas using a h4-accurate three-point formula for ∂x2 often allows the use of fewer grid points and hence faster computations; also it makes the operation of transparent boundaries with Berenger layers considerably more efficient. Finally, the permittivity should always be averaged mesh by mesh.

Limitations of the wide-angle beam propagation method in nonuniform systems

A new approach is proposed that makes clear the basic assumptions in the beam propagation method (BPM) and that leads to an improved formulation free of difficulties with power conservation wide-angle propagation. The new approach allows one to understand the limitations of the wide-angle BPM. Theoretically, arbitrary accuracy could be achieved for weakly guiding systems, even for angles θ ~ 40° from the z axis, provided that a small enough sampling step is used, together with a BPM solver of sufficient order in nonparaxiality. On the contrary, inevitable errors occur with strongly guiding systems because the local-mode expansion of the physical field rapidly involves evanescent local modes, both forward and backward propagating, that cannot be handled in the BPM propagation algorithms. In this case the new formulation is more accurate than the classical BPM for moderate angles only.

Perturbation of a LP mode of an optical fibre by a quasi-degenerate field: a simple formula

A very simple formula is given for the perturbation of a LPon mode of a circular fibre by a quasidegenerate field. Similar formulae, but less simple, are given for higher LPmn modes in circular fibres or for LP modes of non-circular fibres. Some straightforward applications are given.

Reformulation for the beam-propagation method

Splitting scalar fields into a sum of counterpropagating waves leads to formulation of the Helmholtz equation as two coupled parabolic partial differential equations. This rigorous formalism naturally leads to the beam-propagation method (BPM) equation when the backward-propagating field is neglected, but it also permits discussion of various corrections. Two problems specifically are analyzed: wide-angle propagation in near-uniform systems and propagation through lenslike systems, with a strongly discontinuous refractive index. Finally, a BPM modeling is proposed for fiber microlens operation.

Circular Fourier analysis of arbitrarily shaped optical fibers.

Propagation constants of weakly guiding optical fibers with arbitrary shapes can be efficiently calculated through a limited circular Fourier expansion of the wave equation, with a direct numerical integration.

Analysis of tapered mode transformers for semiconductor optical amplifiers

AbstactNumerical data are presented on the operation of field transformers consisting of down-tapered rectangular waveguides. The propagation is analysed through a classical finite difference 3-D BPM method. A new adiabaticity criterion is proposed. Adiabatic and nonadiabatic (linear or exponential) tapers are investigated, allowing one to increase the field radius from 0.5 to 1.5 or 2 μm at λ=1.5 μm, with a coupling loss to a Gaussian beam from 0.1 to 1 dB. For a given overall length, the more efficient field expansion (but also the larger coupling loss) is obtained with an adiabatic design. According to preliminary results on the effect of fabrication imperfections, it seems that linear tapers would be less sensitive to defects than optimized adiabatic tapers.

Radiating normal modes of lossy planar waveguides

The normal modes of an arbitrary, lossey, plane-stratified waveguide symmetric about the propagation plane (i.e. supporting only TE or TM waves) are found from an expansion of its Green’s function. The radiating modes are the transversely bounded fields which results from the scattering of one or two plane waves by the waveguide. Their orthogonality relations are put in a simple form, involving a reflection-like scattering coefficient. A degeneracy occurs in lossless waveguides allowing one to write the orthogonal radiating normal modes under a great variety of forms; some of them lead to symmetric fields in symmetric waveguides. In the lossy case, these degenerate fields split into the “true” normal modes and “improper” modes. The latter may be viewed as analytic continuations of the true modes, they diverge at infinity but they may be useful in some asymptotic evaulations.

Orthogonality and amplitude spectrum of radiation modes along open-boundary waveguides: comment

A comment is made on an orthogonality relation presented in an earlier paper [ J. Opt. Soc. Am.71, 49 ( 1981)].

Finite-difference derivation of the reflectivity at output facets of dielectric waveguides with a highly diverging output beam

Brute force numerical computations are considered for the reflectivity of the output facet of a dielectric waveguide with or without an antireflection coating. Only scalar fields are considered, mainly in two-dimensional systems, through both exact equations and a series expansion, discretized with finite differences. A fair approximation of the series can be obtained with fast Fourier transforms (FFT’s) only, with neither matrix inversion nor diagonalization. Junctions with a highly diverging output beam require a very large computational domain that can be coped with FFT methods only. This is emphasized through the detailed analysis of a three-dimensional example.