Waves can be guided by dielectric rods or slabs and the fields of
these waves occupy the space within and around these dielectric
structures. Especially at optical wavelengths, dielectric fibers are
commonly used to guide waves. In
this section, we develop the properties of waves guided by a planar
sheet of dielectric material. The waves that we find are typical of
those found in integrated optical systems and in the more commonly
used optical fibers of circular cross-section.

A planar version of a dielectric waveguide is pictured in Fig.
13.5.1. A dielectric of thickness *2d* and permittivity _{i} is
surrounded by a dielectric of permittivity * < *_{i}. The latter
might be free space with * = *_{o}. We are interested in how this
structure might be used to guide waves in the *y* direction and will
confine ourselves to fields that are independent of *z*.

** Figure 13.5.1 ** Dielectric slab waveguide.
With a source somewhere to the left (for example an antenna
imbedded in the dielectric), there is reason to expect that there are
fields outside as well as inside the dielectric. We shall look for
field solutions that propagate in the *y* direction and possess
fields solely inside and near the layer. The fields external to the
layer decay to zero in the * x* directions. Like the waves
propagating along waveguides, those guided by this structure have
transverse components that take the form

both inside and outside the dielectric. That is, the fields inside
and outside the dielectric have the same frequency , the same
phase velocity * /k*_{y}, and hence the same wavelength
*2 /k*_{y} in the *y* direction. Of course, whether such fields can
actually exist will be determined by the following analysis.

The classification of two-dimensional fields introduced in Sec.
12.6 is applicable here. The TM and TE fields can be made to
independently satisfy the boundary conditions so that the
resulting modes can be classified as TM or TE.

^{10} Circular
dielectric rods do not support simple TE or TM waves; in that
case, this classification of modes is not possible.

Here we will
confine ourselves to the transverse electric modes. In the exterior
and interior regions, where the permittivities are uniform but
different, it follows from substitution of (1) into (12.6.33)
(Table 12.8.3) that

A guided wave is one that is composed of a nonuniform plane wave
in the exterior regions, decaying in the * x* directions and
propagating with the phase velocity * /k*_{y} in the *y* direction.
In anticipation of this, we have written (2) in terms of the
parameter _{x}, which must then be real and positive. Through the
continuity conditions, the exterior wave must match up to the interior
wave at the dielectric surfaces. The solutions to (3) are sines and
cosines if *k*_{x} is real. In order to match the interior fields onto
the nonuniform plane waves on both sides of the guide, it is necessary
that *k*_{x} be real.

We now set out to find the wave numbers *k*_{y} that not only
satisfy the wave equations in each of the regions, represented by
(2) and (3), but the continuity conditions at the interfaces as well.
The configuration is symmetric about the *x = 0* plane so we can
further divide the modes into those that have even and odd functions
*E*_{z} (x). Thus, with *A* an arbitrary factor, appropriate even
solutions to (2) and (3) are

To simplify the algebra, we have displaced the origin in the exterior
solutions so that just the coefficient, *A*, is obtained when _{z} is
evaluated at the respective interfaces. With a similar objective, the
interior solution has been divided by the constant *cos (k*_{x} d) so
that at the boundaries, _{z} also becomes *A*. In this way,
we have adjusted the interior coefficient so that _{z} is
continuous at the boundaries.

Because this transverse field is the only component of **E**, all
of the continuity conditions on **E** are now satisfied. The
permeabilities of all regions are presumed to be the same, so both
tangential and normal components of **H** must be continuous at the
boundaries. From (12.6.29), the continuity of normal **H** is
guaranteed by the continuity of *E*_{z} in any case. The tangential
field is obtained using (12.6.30).

Substitution of (4) into (5) gives

The assumption that *E*_{z} is even in *x* has as a consequence the
fact that the continuity condition on tangential **H** is satisfied
by the same relation at both boundaries.

Our goal is to determine the propagation constant *k*_{y} for a given
. If we were to
substitute the definitions of _{x} and *k*_{x} into this
expression, we would have this dispersion equation, D(,*k*_{y}),
implicitly relating *k*_{y} to . It is more convenient to
solve for _{x} and *k*_{x} first, and then for *k*_{y}.

Elimination of *k*_{y} between the expressions for _{x} and
*k*_{x} given with (2) and (3) gives a second expression for _{x}
/k_{x}.

The solutions for the values of the normalized transverse wave numbers
*(k*_{x} d) can be pictured as shown in Fig. 13.5.2. Plotted as
functions of *k*_{x} d are the right-hand sides of (7) and (8). The
points of intersection, *k*_{x} d = _{m}, are the desired
solutions. For the frequency used to make Fig. 13.5.2, there are two
solutions. These are designated by even integers because the odd
modes (Prob. 13.5.1) have roots that interleave these even modes.

** Figure 13.5.2 **Graphical solution to (7) and (8).
As the frequency is raised, an additional even TE-guided mode is
found each time the curve representing (8) reaches a new branch of
(7). This happens at frequencies _{c} such that _{x} /k_{x}
= 0 and *k*_{x} d = m /2, where *m = 0, 2, 4, * From (8),

The *m = 0* mode has no cutoff frequency.

To finally determine *k*_{y} from these eigenvalues, the
definition of *k*_{x} given with (3) is used to write

and the dispersion equation takes the graphical form of Fig. 13.5.3.
To make Fig. 13.5.2, we had to specify the ratio of permittivities, so
that ratio is also implicit in Fig. 13.5.3.

** Figure 13.5.3 **Dispersion equation for even TE modes
with _{i} / * = 6.6*.
Features of the dispersion diagram, Fig. 13.5.3, can be gathered
rather simply. Where a mode is just cutoff because * =
*_{c}, _{x} = 0, as can be seen from Fig. 13.5.2. From
(2), we gather that *k*_{y} = _{c} . Thus, at cutoff,
a mode must have a propagation constant *k*_{y} that lies on the
straight broken line to the left, shown in Fig. 13.5.3. At cutoff,
each mode has
a phase velocity equal to that of a plane wave in the medium exterior
to the layer.

In the high-frequency limit, where goes to infinity, we see
from Fig. 13.5.2 that *k*_{x}d approaches the constant
*k*_{x} (m + 1) /2d. That is, in (3), *k*_{x} becomes a
constant even as goes to infinity and it follows that in this high
frequency limit *k*_{y} _{i}.

The physical reasons for this behavior follow from the nature of
the mode pattern as a function of frequency. When _{x}
0, as the frequency approaches cutoff, it follows
from (4) that
the fields extend far into the regions outside of the layer. The wave
approaches an infinite parallel plane wave having a propagation
constant that is hardly affected by the layer. In the opposite
extreme, where goes to infinity, the decay of the external field
is rapid, and a given mode is well confined inside the layer. Again,
the wave assumes the character of an infinite parallel plane wave, but
in this limit, one that propagates with the phase velocity of a plane
wave in a medium with the dielectric constant of the layer.

The distribution of *E*_{z} of the *m = 0* mode at one frequency is
shown in Fig. 13.5.4. As the frequency is raised, each mode becomes
more confined to the layer.

** Figure 13.5.4 ** Distribution of transverse **E** for
TE_{0} mode on dielectric waveguide of Fig. 13.5.1.

#### Demonstration 13.5.1. Microwave Dielectric Guided Waves

In the experiment shown in Fig. 13.5.5, a dielectric slab is
demonstrated to guide microwaves. To assure the excitation of only an
*m = 0* TE-guided wave, but one as well confined to the dielectric as
possible, the frequency is made just under the cutoff frequency
_{c2}. (For a 2 cm thick slab having _{i} /_{o} = 6.6,
this is a frequency just under 6 GHz.) The *m = 0* wave is excited
in the dielectric slab by means of a vertical element at its left
edge. This assures excitation of *E*_{z} while having the symmetry
necessary to avoid excitation of the odd modes.

** Figure 13.5.5 ** Dielectric waveguide
demonstration.
The antenna is mounted at the center of a metal ground plane.
Thus, without the slab, the signal at the receiving antenna (which is
oriented to be sensitive to *E*_{z}) is essentially the same in all
directions perpendicular to the *z* axis. With the slab, a sharply
increased signal in the vicinity of the right edge of the slab gives
qualitative
evidence of the wave guidance. The receiving antenna can also be used
to probe the field decay in the *x* direction and to see that this decay
increases with frequency.

^{11}To make the excitation
independent of *z*, a collinear array of in-phase dipoles could be used
for the excitation. This is not necessary to demonstrate the
qualitative features of the guide.