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
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 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.
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 \hat e_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, \hat e_z also becomes A. In this way, we have adjusted the interior coefficient so that \hat e_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.
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, \ldots 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.
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_xd 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.
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
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.
\footnote11To 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.
