What is implied in the above description is that as solutions of (7.35) and (7.36)
we have a plane wave characterized by a specific direction defined by E 0 and H 0 .
This implies that if we observe the electromagnetic wave at a fixed point, both E and
H oscillate along the mutually perpendicular directions E 0 and H 0 . Hence, we say
that the “electric wave” is polarized in the direction E 0 and that the “magnetic wave”
is polarized in the direction H 0 .
To characterize the polarization of the electromagnetic wave, we introduce
following unit polarization vectors ε e and ε m (with indices e and m related to the
electric and magnetic field, respectively) [3]:
ε e = E 0 =E 0 and ε m = H 0 =H 0 ; H 0 ¼ E 0 =Z,
ð7:66Þ
where E 0 and H 0 are said to be amplitude and may be again complex. We have
ε e  ε m = n:
ð7:67Þ
We call ε e and ε m a unit polarization vector of the electric field and magnetic field,
respectively. As noted above, ε e , ε m , and n constitute a right-handed system in this
order and are mutually perpendicular to one another.
The phase of E in the plane wave (7.58) and that of H in (7.59) are individually
the same on all the points of P. From the wave equations of (7.35) and (7.36),
however, it is unclear whether E and H have the same phase. Suppose that E and H
would have a different phase such that
E = E 0 e
i kÁxÀωt
ð
Þ
and
H = H 0 e
i kÁxÀωtþδ
ð
Þ
¼ H 0 e
iδ e
i kÁxÀωt
ð
Þ
¼ f
H 0 e
i kÁxÀωt
ð
Þ ,
where f
H 0 ¼ H 0 e
iδ
ð
Þis complex and a phase factor e
iδ in the exponent is unknown.
This factor, however, can be set at zero. To show this, let us make qualitative
discussion using Fig. 7.5. Figure 7.5 shows the electric field of the plane wave at
some instant as a function of phase Φ. Suppose that Φ is taken in the direction of n in
Fig. 7.4. Then, from (7.53) we have
Φ ¼ kn Á x À ωt ¼ kn Á ρn À ωt ¼ kρ À ωt,
where ρ is distance from the origin. Also we have
E = E 0 e
i kÁxÀωt
ð
Þ
¼ E 0 ε e e
i kρÀωt
ð
Þ E 0 > 0
ð
Þ:
Suppose furthermore that the phase is measured at t ¼ 0 and that the electric field is
measured along the ε e direction. Then, we have
284
7 Maxwell’s Equations
we have a plane wave characterized by a specific direction defined by E 0 and H 0 .
This implies that if we observe the electromagnetic wave at a fixed point, both E and
H oscillate along the mutually perpendicular directions E 0 and H 0 . Hence, we say
that the “electric wave” is polarized in the direction E 0 and that the “magnetic wave”
is polarized in the direction H 0 .
To characterize the polarization of the electromagnetic wave, we introduce
following unit polarization vectors ε e and ε m (with indices e and m related to the
electric and magnetic field, respectively) [3]:
ε e = E 0 =E 0 and ε m = H 0 =H 0 ; H 0 ¼ E 0 =Z,
ð7:66Þ
where E 0 and H 0 are said to be amplitude and may be again complex. We have
ε e  ε m = n:
ð7:67Þ
We call ε e and ε m a unit polarization vector of the electric field and magnetic field,
respectively. As noted above, ε e , ε m , and n constitute a right-handed system in this
order and are mutually perpendicular to one another.
The phase of E in the plane wave (7.58) and that of H in (7.59) are individually
the same on all the points of P. From the wave equations of (7.35) and (7.36),
however, it is unclear whether E and H have the same phase. Suppose that E and H
would have a different phase such that
E = E 0 e
i kÁxÀωt
ð
Þ
and
H = H 0 e
i kÁxÀωtþδ
ð
Þ
¼ H 0 e
iδ e
i kÁxÀωt
ð
Þ
¼ f
H 0 e
i kÁxÀωt
ð
Þ ,
where f
H 0 ¼ H 0 e
iδ
ð
Þis complex and a phase factor e
iδ in the exponent is unknown.
This factor, however, can be set at zero. To show this, let us make qualitative
discussion using Fig. 7.5. Figure 7.5 shows the electric field of the plane wave at
some instant as a function of phase Φ. Suppose that Φ is taken in the direction of n in
Fig. 7.4. Then, from (7.53) we have
Φ ¼ kn Á x À ωt ¼ kn Á ρn À ωt ¼ kρ À ωt,
where ρ is distance from the origin. Also we have
E = E 0 e
i kÁxÀωt
ð
Þ
¼ E 0 ε e e
i kρÀωt
ð
Þ E 0 > 0
ð
Þ:
Suppose furthermore that the phase is measured at t ¼ 0 and that the electric field is
measured along the ε e direction. Then, we have
284
7 Maxwell’s Equations