In this lecture we will analyze the mode
structure of the waveguide consisting of
the dielectric core sandwich between
two metal cladding layers. Let's start.
Let's consider a parallel plate waveguide
with a dielectric core and metal claddings.
This waveguide has two interfaces
separating the core from the claddings,
and each of these interfaces support
propagating surface plasmon polariton.
Therefore, we could assume that such kind
of a waveguide supports two types of the
modes: the first type of the eigenmodes is
just a conventional waveguide mode,
forming due to the total internal
reflection from the cladding layers, and
the second type of the mode is the
surface plasmon polariton mode, forming
due to the constructive and destructive
interference of surface plasmon
polaritons localized at the opposite
interfaces. Let's try to solve this
problem rigorously, it means let's try to
find the dispersion for each of the
eigenmodes in this waveguide, and let's
try to find the field distribution for
each of the eigenmodes. Here we will
consider only the TM waves, for the TM waves we
have only two non-zero components of
the electric field: X component and Z component,
and Y component of the magnetic field,
which is perpendicular to the plane of
the screen, so the method how is possible to
solve this problem is well known, we
could find the general solution in each
of the media and then we can apply
the boundary conditions at the
interfaces and obtain the dispersion
equation. But actually, there is another,
more simple method how it is possible to
obtain the dispersion equation. The
method is the following: if we know
the reflection coefficients R1 and
R2, and we know that these
coefficients are given by the Fresnel
equations, after the double reflection
the wave should come to, there, to this
interface
in phase with itself. It means that the
following equations should be fulfilled,
where k_x, this factor, this exponential
factor, comes through the phase delay
because of the propagation of the wave
from this interface
to this and back. In the case when
these coefficients are equal to each
other, this is our case, we could just
rewrite this equation in the following
form: here we have two brackets, and one
bracket corresponds to the symmetric mode
and one bracket corresponds to the
anti-symmetric mode.
It's important to know that in
our case we have two components of the
electric field, as I said, and if the Z
component of the electric field for the TM wave
is an even function, it means that the X
component will be an odd function. I will
classify the modes in this waveguide
with respect to the symmetry of the Z
component of the electric field. It means
that under symmetric mode I will apply the
mode with a symmetric distribution of
the Z component of the electric field,
and it's possible to rewrite this
dispersion equation just substituting
the expression for the reflection
coefficient of the bracket, and we will
obtain the following dispersion equation
for symmetric mode. This dispersion
equations could be rewritten
in the following form. It means
that we need just to solve two of these
equations to obtain the dispersion of
the eigenmodes. Surprisingly, but in spite
of the fact that these dispersion
equations are transcendental equations,
they could be solved analytically in
parametric form. We need just to solve
these equations,
using the k_x^d as a parameter. But you can
do it by yourself as a simple exercise.
This is the solution of these dispersion
equations for the symmetric and antisymmetric
modes. One can see that the spectrum of
the waveguide indeed consists of two
types of the eigenmodes:
the first type of the eigenmodes is just
conventional waveguide modes, each of
them has a frequency cutoff, and at
higher frequencies the
dispersion tends to the light line.
The second type of the eigenmodes is
the surface plasmon polariton mode,
the symmetric and the anti-symmetric one. Here
we see that for the symmetric mode there
is a frequency cutoff, but for the
anti-symmetric mode there is no
frequency cutoff, it means that this mode
can propagate in this waveguide at
arbitrary low frequencies. It's very
important for some practical
applications that we will discuss further.
And another important moment is that in this
region, in the vicinity of the surface
plasmon polariton resonance, these modes,
symmetric and anti-symmetric, are almost
degenerated. Of course, they are degenerated
because in this case surface plasmon
polaritons are localized on opposite
interfaces and do not interact with each
other. It is why we have such kind of
degeneracy.
Now let's consider a thin metal-insulator-
metal waveguide. In a thin waveguide we
could expect the strong coupling between
the plasmons propagating along the
opposite interfaces. Indeed, here we see
that in this waveguide we have only
three eigenmodes,
so it's one conventional waveguide mode,
the antisymmetric one, and two surface
plasmon polariton modes, and in contrast
to the previous case we see that we have
a strong repulsion of the frequencies. It
happens because of the strong
coupling which occurs because of the
our waveguide core is very thin. Another
interesting fact is that for the symmetric
mode we have a negative dispersion,
and the origin of the negative dispersion is
very simple: for the symmetric mode the
main part of the energy is contained inside
the metal claddings.
As we know, in the metal the energy
propagates against the propagation
constant, it is why for the symmetric
mode we have a negative dispersion.
And this is this summary of the lecture.