being propagated in the direction of the positive z-axis; δ is a phase difference. The
total electric field E is described as the superposition of E 1 and E 2 such that
E = E 1 þ E 2 = E 1 e 1 e
i kzÀωt
ð
Þ
þ E 2 e 2 e
i kzÀωtþδ
ð
Þ
:
ð7:69Þ
Note that we usually discuss the polarization characteristics of electromagnetic
wave only by considering electric waves. We emphasize that an electric wave and
concomitant magnetic wave share the same phase in a uniform and infinite dielectric
media. A reason why the electric wave represents an electromagnetic wave is partly
because optical application is mostly made in a nonmagnetic substance such as glass,
water, plastics, and most of semiconductors.
Let us view temporal change of E at a fixed point x = 0; x ¼ y ¼ z ¼ 0. Then,
taking a real part of (7.69), x- and y-components of E; i.e., E x and E y are expressed as
E x ¼ E 1 cos Àωt
ð
Þand E y ¼ E 2 cos Àωt þ δ
ð
Þ :
ð7:70Þ
First, let us briefly think of the case where δ ¼ 0. Eliminating t, we have
E y ¼
E 2
E 1
E x :
ð7:71Þ
This is an equation of a straight line. The resulting electric field E is called a linearly
polarized light accordingly. That is, when we are observing the electric field of the
relevant light at the origin, the field is oscillating along the straight line described by
(7.71) with the origin centrally located of the oscillating field. If δ ¼ π, we have
E y ¼ À
E 2
E 1
E x :
This gives a straight line as well. Therefore, if we wish to seek the relationship
between E x and E y , it suffices to examine it as a function of δ in a region of À
π
2
δ
π
2 .
(i) Case I: E 1 6 ¼ E 2 .
Let us consider the case where δ 6 ¼ 0 in (7.70). Rewriting the second equation of
(7.70) and inserting the first equation into it so that we can eliminate t, we have
E y ¼ E 2 cos ωt cos δ þ sin ωt sin δ
ð
Þ ¼ E 2 cos δ
E x
E 1
Æ sin δ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
E x
2
E
2
1
s
!
:
Rearranging terms of the above equation, we have
286
7 Maxwell’s Equations
total electric field E is described as the superposition of E 1 and E 2 such that
E = E 1 þ E 2 = E 1 e 1 e
i kzÀωt
ð
Þ
þ E 2 e 2 e
i kzÀωtþδ
ð
Þ
:
ð7:69Þ
Note that we usually discuss the polarization characteristics of electromagnetic
wave only by considering electric waves. We emphasize that an electric wave and
concomitant magnetic wave share the same phase in a uniform and infinite dielectric
media. A reason why the electric wave represents an electromagnetic wave is partly
because optical application is mostly made in a nonmagnetic substance such as glass,
water, plastics, and most of semiconductors.
Let us view temporal change of E at a fixed point x = 0; x ¼ y ¼ z ¼ 0. Then,
taking a real part of (7.69), x- and y-components of E; i.e., E x and E y are expressed as
E x ¼ E 1 cos Àωt
ð
Þand E y ¼ E 2 cos Àωt þ δ
ð
Þ :
ð7:70Þ
First, let us briefly think of the case where δ ¼ 0. Eliminating t, we have
E y ¼
E 2
E 1
E x :
ð7:71Þ
This is an equation of a straight line. The resulting electric field E is called a linearly
polarized light accordingly. That is, when we are observing the electric field of the
relevant light at the origin, the field is oscillating along the straight line described by
(7.71) with the origin centrally located of the oscillating field. If δ ¼ π, we have
E y ¼ À
E 2
E 1
E x :
This gives a straight line as well. Therefore, if we wish to seek the relationship
between E x and E y , it suffices to examine it as a function of δ in a region of À
π
2
δ
π
2 .
(i) Case I: E 1 6 ¼ E 2 .
Let us consider the case where δ 6 ¼ 0 in (7.70). Rewriting the second equation of
(7.70) and inserting the first equation into it so that we can eliminate t, we have
E y ¼ E 2 cos ωt cos δ þ sin ωt sin δ
ð
Þ ¼ E 2 cos δ
E x
E 1
Æ sin δ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
E x
2
E
2
1
s
!
:
Rearranging terms of the above equation, we have
286
7 Maxwell’s Equations