�
� �
The rotation angle is then given by
χ ab = 2
1
z b
p
−1 B dz.
(2.157)
za
This gives rise to the following useful transformations:
¯
= v v
u u
¯
� �
� �
1
�
−2 B
2
u ¯ u = v v + 2 p
−1 B i (¯ v − ¯
� ) +
1
4
p
¯
¯
v
v v
v v
i (¯ u
�
u u
=
v
�
v v
v v.
u − ¯
� )
i (¯ v − ¯
� ) + p
−1 B ¯
(2.158)
Substituting these into (2.151) we obtain the modified refractive
index in the rotated coordinates as
m = m 0 + m 2 + m 4 + . . .

= p

1
� �
+ 2 p v ¯ v
+ − 4
1 p
−1 Φ
�� (1 + Φ) −
1
8
p
−1 B
2 v v
¯
1 −3 Φ
�� 2
1
+ [
1 p
−1 Φ
IV (1 + Φ) − p
+ p
−1 B B
��
64
32
32
1
1
2 2
− p
−3 B
4 − p
−3 Φ
�� (1 + Φ) B
2 ] ¯
v v
128
32
−
1
1 −1 B
2
� �
+
8
p
−1 Φ
�� (1 + Φ) − 16 p
v v ¯ v
¯ v
−
1
� 2 � 2
+ 8 p v ¯ v
1
1 B
��
+ −
1 p
−2 B
3 − 16 p
−2 Φ
�� (1 + Φ) B +
32
32
v
v v
· i (¯
� v − ¯
� ) ¯
v v
+ − 8
1 B i (¯ v v − v v
¯
� ) ¯
v v
−
1 −1 B
2
�
2
32
[ i ( ¯ v − ¯
+
p
v
v v
� ) ]
+ . . . .
(2.159)
The paraxial term is given in the rotated system by
1
� �
−1 B
2
m 2 = p v ¯ v + −
1 p
−1 Φ
�� (1 + Φ) −
1 p
¯
(2.160)
vv.
2
4
8
The paraxial approximation to (2.104) is then
∂m 2
d ∂m 2
−
�
= 0.
(2.161)
∂v ¯
dz ∂v ¯
59
2.5. Axial symmetry