WAVE PROPAGATION IN OPTICAL WAVEGUIDES

This research started with a collaboration between myself and F. Santosa (IMA, University of Minnesota, Minneapolis) and continued when a student of mine, G. Ciraolo, and a student of Santosa’s, O. Aleksandrov, merged their overlapping results on this problem.

It is considered the Helmholtz or (time-harmonic wave) equation in the Euclidean 3-dimensional space:

D u+k² n(x,y,z)² u= f

Here, k denotes the wave number, f is a source term, and n the (non-homogeneous) index of refraction of the medium. A good model for an infinite optical fiber takes place when the index of refraction has the following form:

n(x,y,z)=n_cl if (x,y,z)ÎW´ (-¥,+¥),

n(x,y,z)=n_co(x,y) if (x,y,z)ÏW´ (-¥,+¥),

where n_cl is a positive constant, n_co(x,y) is a function that we take bounded below by , and W is a bounded domain in the (x,y)-plane. The subscripts cl and co indicate respectively the fiber’s cladding, where the index of refraction is assumed constant, and the fiber’s core, glasslike, where most of the light energy propagates.

In the physical situation, roughly speaking, the energy released by the source splits up into 2 parts: one is trapped inside the core (guided modes) while the rest gets dissipated outside. Our aim is to provide a mathematically rigorous analysis of some facts present in the specialized literature that could constitute a firm groundwork for further research in the field.

In [1], we considered a 2-dimensional version of this problem, in which the fiber is described by a diffraction coefficient n(x,z) which is constant outside the strip [-h,h] ´ (-¥,+¥) and depends only on the variable x inside the strip. By taking advantage of the symmetries of the problem, we separate variables and, by using the well-known Titchmarsh’s theory on series expansions of eigenfunctions of certain Sturm-Liouville problems, we obtained a formula for the Green’s function of the problem. Such a formula involves a Lebesgue-Stieltjes integral on a suitable spectrum and makes it clear how the energy of a source splits up into 2 parts: one corresponds to a finite number of eigenvalues (the guided modes), propagates along the fiber and is essentially confined in it, since it vanishes exponentially outside, the other one corresponds to a continuum in which, in turn, we can distinguish radiating modes (non vanishing and oscillating modes that propagate outside the fiber) and evanescent modes (which rapidly decay along the fiber).

An application of this formula is given in [2].

O. Aleksandrov and G. Ciraolo, the former working for his Ph. D. thesis and the latter for his tesi di laurea, took up the task of extending the results obtained in [1] to the 3-dimensional situation in which W is a disk and n_co only depends on the distance from the fiber’s axis z. The computations needed to treat this case are sensibly more difficult, essentially because the relevant Sturm-Liouville systems to consider possess extra singularities and a more refined use of Titchmarsh’s theory must be made.

However, the 3-modes structure of the Green’s function is confirmed if we except a new kind of mode which occurs when the last guided mode in the discrete spectrum coalesces with the continuous spectrum. Such new mode is not radiating or evanescent, because vanishes outside the fiber and does not along the fiber, but it cannot be considered a guided mode either because it vanishes polynomially outside the fiber.

The results obtained by O. Aleksandrov and G. Ciraolo are summarized in [3], for the theoretical part and in [4] for the numerical one.

The pictures were produced by G. Ciraolo by using their formula. They show the level surfaces (or xz-sections of them) of the Green’s function with singularity located in various positions.