Types of Modes in Planar Optical Waveguides

Types of Modes in Planar Optical Waveguides

Light behaviour in an optical waveguide can initially be analysed by examining the
case of an asymmetric planar waveguide from the point of view of geometric optics
(ray optics).
Let us consider the planar waveguide depicted in Figure 3.9, where we have assumed
that the refractive index of the film nf is higher than the refractive index corresponding
to the substrate ns and the upper cover nc. In addition, we assume the usual situation in
which the relation ns > nc is fulfilled. In this way, the critical angles that define total
internal reflection for the cover–film interface (θ1c) and the film–substrate boundary
(θ2c) are determined by:
θ1c = sin−1(nc/nf ) (3.1)
θ2c = sin−1(ns/nf )


In addition, as we have nf > ns > nc, it follows that the critical angles fulfil the
relation θ2c > θ1c. If now we fix our attention to the propagating angle θ of the light
inside the film (Figure 3.9), three situations can be distinguished:
(i) θ < θ1c. In this case, if the ray propagates with internal angles θ lower than the
critical angle corresponding to the film–cover interface θ1c, the light penetrates
the cover, as well as the substrate, because θ2c > θ1c. Thus, the radiation is not
confined to the film, but travels in the three regions. This situation corresponds
to radiation modes, because the light radiates to the cover and the substrate
(Figure 3.10).
(ii) θ1c < θ < θ2c. Light travelling in these circumstances is totally reflected at the
film–cover interface, thus it cannot penetrate the cover region. Nevertheless, the
radiation can still penetrate the substrate, and therefore it corresponds to substrate
radiation modes, or in short, substrate modes (Figure 3.11).
(iii) θ2c < θ < π/2. In this situation, the ray will suffer total internal reflection at the
upper and lower interfaces, and thus the radiation is totally confined and cannot
escape the film. This situation corresponds to a guided mode (Figure 3.12), and
is the most relevant case in integrated optics.

Ray optics analysis of guided modes Although the light propagation in waveguide
structures should be analysed by a rigorous electromagnetic wave treatment, an analysis
based on optical rays is not only more intuitive, but in addition its solution to the
problem coincides with that supplied by the more rigorous wave treatment. The optical
ray approach of the guided modes in planar optical waveguides consists of studying a
ray inside the film moving on a zig-zag path. The first condition which a ray of light
must fulfil in order to be confined in the film region is that the angle of incidence at
the upper and lower interfaces must be higher that the critical angles defined by the
cover–film and film–substrate boundaries (Figure 3.13), that is, θ < θ1c, θ2c.
In a round trip inside the film, the ray suffers a transversal phase shift that depends
on the film thickness, and also additional phase shifts due to total internal reflection
at the two boundaries. The condition for a guided mode is established on the basis
of constructive interference, which implies that the total transversal phase shift in a
complete round trip should be an integral number of 2π. Only a discrete number of
angles fulfils that condition, and these will correspond to the propagation angles of
guided modes.


This constitutes a link between the ray picture of the guided modes, characterised by
its propagation angle θm, and the electromagnetic wave treatment that considers the
mode characterised by its propagation constant βm.
The ray optic approach that we have carried out can be used for the qualitative
description of light behaviour in an optical waveguide, to establish the types of mode
that can be found in such structures, to calculate the number of guided modes that
support a waveguide, and to determine its propagation constants. Nevertheless, for
many applications it is essential to know the electric field distribution of the radiation
within the waveguide structure, and this method does not provide such information. If
one wants to determine the optical fields or the intensity distribution associated with
the light propagation in waveguide structures, it becomes necessary to invoke a more
rigorous formalism, based on the electromagnetic theory of the light, as explained in
Chapter 2. Therefore, the problem should start from Maxwell’s equations applied to the
electromagnetic fields in a given structure, which defines the waveguide; the solutions
for the fields will correspond to the propagation modes.