## A path integral approach to the coupled-mode equations with specific reference to optical waveguides

##### Date

2009-03

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Stellenbosch : University of Stellenbosch

##### Abstract

The propagation of electromagnetic radiation in homogeneous or periodically modulated media can
be described by the coupled mode equations. The aim of this study was to derive analytical expressions
modeling the solutions of the coupled-mode equations, as alternative to the generally used numerical
and transfer-matrix methods. The path integral formalism was applied to the coupled-mode equations.
This approach involved deriving a path integral from which a generating functional was obtained. From
the generating functional a Green’s function, or propagator, describing the nature of mode propagation
was extracted. Initially a Green’s function was derived for the propagation of modes having position
independent coupling coefficients. This corresponds to modes propagating in a homogeneous medium
or in a uniform grating formed by a periodic variation of the index of refraction along the direction of
propagation. This was followed by the derivation of a Green’s function for the propagation of modes having
position dependent coupling coefficients with the aid of perturbation theory. This models propagation
through a nonuniform inhomogeneous medium, specifically a modulated grating.
The propagator method was initially tested for the case of propagation in an arbitrary homogeneous
medium. In doing so three separate cases were considered namely the copropagation of two modes in
the forward and backward directions followed by the counter propagation of the two modes. These more
trivial cases were used as examples to develop a rigorous mathematical formalism for this approach. The
results were favourable in that the propagator’s results compared well with analytical and numerical
solutions.
The propagator method was then tested for mode propagation in a periodically perturbed waveguide.
This corresponds to the relevant application of mode propagation in uniform gratings in optical fibres.
Here two case were investigated. The first scenario was that of the copropagation of two modes in a long
period transmission grating. The results achieved compared well with numerical results and analytical
solutions. The second scenario was the counter propagation of two modes in a short period reflection
grating, specifically a Bragg grating. The results compared well with numerical results and analytical
solutions. In both cases it was shown that the propagator accurately predicts many of the spectral
properties of these uniform gratings.
Finally the propagator method was applied to a nonuniform grating, that is a grating for which the
uniform periodicity is modulated - in this case by a raised-cosine function. The result of this modulation
is position dependent coupling coefficients necessitating the use of the Green’s function derived using
perturbation theory. The results, although physically sensible and qualitatively correct, did not compare
well to the numerical solution or the well established transfer-matrix method on a quantitative level at
wavelengths approaching the design wavelength of the grating. This can be explained by the breakdown
of the assumptions of first order perturbation theory under these conditions.

##### Description

MSc

Thesis (MSc (Physics))--University of Stellenbosch, 2009.

Thesis (MSc (Physics))--University of Stellenbosch, 2009.

##### Keywords

Coupled-mode equations, Optical mode propagation, Dissertations -- Physics, Theses -- Physics