We have two cases about a geometry of the fields (see Fig. 8.4). The first case is
that all E i , E r , and E t are directed in the same direction (i.e., the positive direction of
the x-axis); see Fig. 8.4a. Another case is that although E i and E t are directed in the
same direction, E r is reversed (Fig. 8.4b). In this case, we define E r as negative.
Notice that E i and E t are always directed in the same direction and that E r is directed
either in the same direction or in the opposite direction according to the nature of the
dielectrics. The situation will be discussed soon.
Meanwhile, unit polarization vectors of the magnetic fields are determined by
(7.67) for the incident, reflected, and transmitted waves. In Fig. 8.4, the magnetic
fields are polarized along the y-axis (i.e., perpendicular to the plane of paper). The
magnetic fields H i and H t are always directed to the same direction as in the case of
the electric fields. On the other hand, if the phase of E r is conserved, the direction of
H r is reversed and vice versa. This converse relationship with respect to the electric
and magnetic fields results solely from the requirement that E, H, and the propagation unit vector n of light must constitute a right-handed system in this order. Notice
that n is reversed upon reflection.
Next, let us consider an oblique incidence. With the oblique incidence, electromagnetic waves are classified into two special categories, i.e., transverse electric
(TE) waves (or modes) or transverse magnetic (TM) waves (or modes). The TE wave
is characterized by the electric field that is perpendicular to the incidence plane,
whereas the TM wave is characterized by the magnetic field that is perpendicular to
the incidence plane. Here the incidence plane is a plane that is formed by the
propagation direction of the incident light and the normal to the interface of the
two dielectrics. Since E, H, and n form a right-handed system, in the TE wave H lies
on the incidence plane. For the same reason, in the TM wave E lies on the incidence
plane.
In a general case where a field is polarized in an arbitrary direction, that field can
be formed by superimposing two fields corresponding to the TE and TM waves. In
other words, if we take an arbitrary field E, it can be decomposed into a component
having a unit polarization vector directed perpendicular to the incidence plane and
another component having the polarization vector that lies on the incidence plane.
These two components are orthogonal to each other.
Example 8.1: TE Wave In Fig. 8.5 we depict the geometry of oblique incidence of
a TE wave. The xy-plane defines the interface of the two dielectrics and t of (8.9) lies
on that plane. The zx-plane defines the incidence plane. In this case, E is polarized
along the y-axis with H polarized in the zx-plane. That is, regarding E, we choose
polarization direction ε i , ε r , and ε t of the electric field as e 2 (a unit vector toward the
positive direction of the y-axis that is perpendicular to the plane of paper). In Fig. 8.5,
the polarization direction of the electric field is denoted by a symbol ⨂. Therefore,
we have
e 2 Á ε i = e 2 Á ε r = e 2 Á ε t = 1:
ð8:40Þ
304
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media
that all E i , E r , and E t are directed in the same direction (i.e., the positive direction of
the x-axis); see Fig. 8.4a. Another case is that although E i and E t are directed in the
same direction, E r is reversed (Fig. 8.4b). In this case, we define E r as negative.
Notice that E i and E t are always directed in the same direction and that E r is directed
either in the same direction or in the opposite direction according to the nature of the
dielectrics. The situation will be discussed soon.
Meanwhile, unit polarization vectors of the magnetic fields are determined by
(7.67) for the incident, reflected, and transmitted waves. In Fig. 8.4, the magnetic
fields are polarized along the y-axis (i.e., perpendicular to the plane of paper). The
magnetic fields H i and H t are always directed to the same direction as in the case of
the electric fields. On the other hand, if the phase of E r is conserved, the direction of
H r is reversed and vice versa. This converse relationship with respect to the electric
and magnetic fields results solely from the requirement that E, H, and the propagation unit vector n of light must constitute a right-handed system in this order. Notice
that n is reversed upon reflection.
Next, let us consider an oblique incidence. With the oblique incidence, electromagnetic waves are classified into two special categories, i.e., transverse electric
(TE) waves (or modes) or transverse magnetic (TM) waves (or modes). The TE wave
is characterized by the electric field that is perpendicular to the incidence plane,
whereas the TM wave is characterized by the magnetic field that is perpendicular to
the incidence plane. Here the incidence plane is a plane that is formed by the
propagation direction of the incident light and the normal to the interface of the
two dielectrics. Since E, H, and n form a right-handed system, in the TE wave H lies
on the incidence plane. For the same reason, in the TM wave E lies on the incidence
plane.
In a general case where a field is polarized in an arbitrary direction, that field can
be formed by superimposing two fields corresponding to the TE and TM waves. In
other words, if we take an arbitrary field E, it can be decomposed into a component
having a unit polarization vector directed perpendicular to the incidence plane and
another component having the polarization vector that lies on the incidence plane.
These two components are orthogonal to each other.
Example 8.1: TE Wave In Fig. 8.5 we depict the geometry of oblique incidence of
a TE wave. The xy-plane defines the interface of the two dielectrics and t of (8.9) lies
on that plane. The zx-plane defines the incidence plane. In this case, E is polarized
along the y-axis with H polarized in the zx-plane. That is, regarding E, we choose
polarization direction ε i , ε r , and ε t of the electric field as e 2 (a unit vector toward the
positive direction of the y-axis that is perpendicular to the plane of paper). In Fig. 8.5,
the polarization direction of the electric field is denoted by a symbol ⨂. Therefore,
we have
e 2 Á ε i = e 2 Á ε r = e 2 Á ε t = 1:
ð8:40Þ
304
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media