2.5.3 Series solution for the general ray equation
We now seek a general solution to the exact ray equation (2.104).
This must include all rays, including meridional and skew rays.
Because the exact trajectory equations (2.134, 2.136) are nonlinear, they cannot be solved in closed form. Consequently, we seek
an approximate solution by series expansion [33].
Recalling the modified refractive index for the general curvilinear
axis,
√
m = p 1 + r � 2 + r 2 θ � 2 − rθ
� A θ ,
(2.147)
where the scalar kinetic momentum p(r, z) is given by (2.123) and
the magnetic vector potential A θ is given by (2.130).
We define a complex transverse coordinate
iθ
−iθ
u = X + i Y = r e ,
u ¯ = X − i Y = r e ,
(2.148)
where X(z) and Y (z) are Cartesian coordinates in a transverse
plane at axial coordinate z. It follows that
2 θ
�
i (¯ u
� u − uu ¯
� ) = 2(X Y
� − X
� Y ) = 2 r ,
(2.149)
and
√
√
�
� 2 � 2
�
� −
1
1 + r � 2 + r 2 θ � 2 = 1 + ¯
u u � = 1 + 2
1 u ¯ u
8
u ¯ u + . . . .
(2.150)
We can write a power series expansion for the refractive index,
making use of the axial symmetry of the scalar kinetic momentum
p(r, z) and the magnetic vector potential A θ (r, z) in (2.123, 2.130)
as follows:
m = m 0 + m 2 + m 4 + . . .

= p

+ −
1
4
p
−1 Φ
�� (1 + Φ) uu ¯
1
� �
+ 2 p u ¯ u
57
2.5. Axial symmetry