Radiation modes

Radiation modes

Up to now we have examined the solution of the wave equation for

planar waveguides in terms of guided modes, where the radiation is mainly confined
within the film, with decaying solution at both the cover and substrate regions in
the form of evanescent waves. In this case, the mode effective index was restricted
between the refractive index of the film and that of the substrate. Nevertheless, the wave
equation, for both TE and TM polarisation light, also admits solutions for effective
indices lower than ns . In this case, we are dealing with radiation modes, where the
light is no longer confined to the film, but can “leak” to adjacent regions, losing the
light power inside the film core as the wave propagates along the waveguide. For this
reason, these types of solutions are often called leaky modes.
Following the discussion of paragraph 3.3, outlined in Figure 3.16, for effective
refractive index values lower than ns but higher than nc (nc < N < ns , k0nc < β <
k0ns ), the solutions in the film and substrate regions are in the form of oscillatory
functions, while the behaviour of the fields in the cover region is in the form of
exponential decay. This situation corresponds to substrate radiation modes, where the
light is not confined to the film region, but also spreads out to the substrate, as can be
seen in Figure 3.22. In addition, the solutions for leaky substrate modes are not discrete,
but instead the wave equation for substrate modes admits an infinite number of solutions
for continuous propagation constant values β (or effective refractive indices N).
Finally, if the mode effective refractive index N is lower than nc (N <nc,β < k0nc)
the solution for the modal fields in the three regions is in the form of sinusoidal
functions. In this case the field pattern corresponds to a radiation mode, where the
light cannot be confined in the film but leaks to the cover and substrate regions, as can
be seen in Figure 3.23. Also, as in the case of substrate modes, there exists a continuous
and infinite number of values for the propagation constant of radiation modes, with an
infinite number of solutions for the electromagnetic field distribution.