Concomitantly, we adopt Neumann conditions as the boundary conditions (see
Sects. 1.3 and 8.3). Regardless of the difference in the boundary conditions, however, discussion including (8.149) to (8.158) applies to the analysis of TM waves.
Once H x is determined, E y and E z can be determined as well from (8.134) and
(8.135).
8.7.2 Total Internal Reflection and Evanescent Waves
If a slab waveguide shaped by a dielectric is sandwiched by a couple of another
dielectric (Fig. 8.10b), the situation differs from a metal waveguide (Fig. 8.10a) we
encountered in Sect. 8.7.1. Suppose in Fig. 8.10b that the former dielectric D1 of a
refractive index n 1 is sandwiched by the latter dielectric D2 of a refractive index n 2 .
Suppose that an electromagnetic wave is being propagated from D1 toward D2.
Then, we must have
n 1 > n 2
ð8:159Þ
so that the total internal reflection can take place at the interface of D1 and D2. In this
case, the dielectrics D1 and D2 act as a core layer and a clad layer, respectively.
The biggest difference between the present waveguide and the previous one is
that unlike the previous case, the total internal reflection occurs in the present case.
Concomitantly, an evanescent wave is present in the clad layer very close to the
interface.
First, let us estimate the conditions that are satisfied so that an electromagnetic
wave can be propagated within a waveguide. Figure 8.13 depicts a cross-section of
the waveguide where the light is propagated in the direction of k. In Fig. 8.13,
suppose that we have a normal N to the plane of paper at P. Then, N and a straight
line XY shape a plane NXY. Also suppose that a dielectric fills a semi-infinite space
situated below NXY. Further suppose that there is another virtual plane N’X’Y’ that
is parallel with NXY as shown. Here N
0 is parallel to N. The separation of the two
parallel plane is d. We need the virtual plane N’X’Y
0 just to estimate an optical path
difference (or phase difference, more specifically) between two waves, i.e., a propagating wave and a reflected wave.
Let n be a unit vector in the direction of k; i.e.,
n ¼ k= j k j¼ k=k:
ð8:160Þ
Then the electromagnetic wave is described as
E = E 0 e
i kÁxÀωt
ð
Þ
¼ E 0 e
i knÁxÀωt
ð
Þ
:
ð8:161Þ
Suppose that we take a coordinate system such that
8.7 Waveguide Applications
327
Sects. 1.3 and 8.3). Regardless of the difference in the boundary conditions, however, discussion including (8.149) to (8.158) applies to the analysis of TM waves.
Once H x is determined, E y and E z can be determined as well from (8.134) and
(8.135).
8.7.2 Total Internal Reflection and Evanescent Waves
If a slab waveguide shaped by a dielectric is sandwiched by a couple of another
dielectric (Fig. 8.10b), the situation differs from a metal waveguide (Fig. 8.10a) we
encountered in Sect. 8.7.1. Suppose in Fig. 8.10b that the former dielectric D1 of a
refractive index n 1 is sandwiched by the latter dielectric D2 of a refractive index n 2 .
Suppose that an electromagnetic wave is being propagated from D1 toward D2.
Then, we must have
n 1 > n 2
ð8:159Þ
so that the total internal reflection can take place at the interface of D1 and D2. In this
case, the dielectrics D1 and D2 act as a core layer and a clad layer, respectively.
The biggest difference between the present waveguide and the previous one is
that unlike the previous case, the total internal reflection occurs in the present case.
Concomitantly, an evanescent wave is present in the clad layer very close to the
interface.
First, let us estimate the conditions that are satisfied so that an electromagnetic
wave can be propagated within a waveguide. Figure 8.13 depicts a cross-section of
the waveguide where the light is propagated in the direction of k. In Fig. 8.13,
suppose that we have a normal N to the plane of paper at P. Then, N and a straight
line XY shape a plane NXY. Also suppose that a dielectric fills a semi-infinite space
situated below NXY. Further suppose that there is another virtual plane N’X’Y’ that
is parallel with NXY as shown. Here N
0 is parallel to N. The separation of the two
parallel plane is d. We need the virtual plane N’X’Y
0 just to estimate an optical path
difference (or phase difference, more specifically) between two waves, i.e., a propagating wave and a reflected wave.
Let n be a unit vector in the direction of k; i.e.,
n ¼ k= j k j¼ k=k:
ð8:160Þ
Then the electromagnetic wave is described as
E = E 0 e
i kÁxÀωt
ð
Þ
¼ E 0 e
i knÁxÀωt
ð
Þ
:
ð8:161Þ
Suppose that we take a coordinate system such that
8.7 Waveguide Applications
327