¼ E x E y
À
Á
P
1 þ cos δ
0
0
1À cos δ
P
À1
E x
E y
¼ E
2
1 sin
2
δ:
ð7:86Þ
Here, let us define new coordinates such that
u
v
P
À1
E x
E y
:
ð7:87Þ
This coordinate transformation corresponds to the transformation of basis vectors
(e 1 e 2 ) such that
e 1 e 2
ð
Þ
E x
E y
¼ e 1 e 2
ð
ÞPP
À1
E x
E y
¼ e
0
1 e
0
2
À
Á u
v
,
ð7:88Þ
where new basis vectors e
0
1 e
0
2
À
Á
are given by
e
0
1 e
0
2
À
Á ¼ e 1 e 2
ð
ÞP ¼
1
ffiffi ffi
2
p e 1 À
1
ffiffi ffi
2
p e 2
1
ffiffi ffi
2
p e 1 þ
1
ffiffi ffi
2
p e 2
:
ð7:89Þ
The coordinate system is depicted in Fig. 7.7 along with the basis vectors. The
relevant discussion will again appear in Part III.
Substituting (7.87) for (7.86) and rearranging terms, we get
u
2
E
2
1 1 À cos δ
ð
Þ
þ
v
2
E
2
1 1 þ cos δ
ð
Þ
¼ 1:
ð7:90Þ
Equation (7.90) indicates that a major axis and minor axis are E 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ cos δ
p
and
E 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À cos δ
p
, respectively. When δ ¼ Æ π/2, (7.90) becomes
Fig. 7.7 Relationship
between the basis vectors
(e 1 e 2 ) and e
0
1 e
0
2
À
Á
in the
case of E 1 ¼ E 2 ; see text
7.4 Superposition of Two Electromagnetic Waves
291