where we now proceed to solve for the coefficients b j . We define a
function B(z) as the magnetic field on axis,
B(z) ≡ B z (0, z) = 2b 1
(2.129)
leading to
1 B
�� 3
1 B
IV 5
A θ (r, z) =
1 Br −
r +
r + . . . .
(2.130)
2
16
384
The modified refractive index m is written as
ds
m = n dz
ds
= (p − A · ˆ s) dz
√
v θ A θ ds
= p 1 + r � 2 + r 2 θ � 2 − ds/dt dz
√
= p 1 + r � 2 + r 2 θ � 2 − rθ
� A θ .
(2.131)
Euler-Lagrange equation for angular coordinate is (2.104, 2.131)
∂m
d ∂m
∂θ
− dz ∂θ � = 0,
(2.132)
where, because of axial symmetry,
∂m = 0.
(2.133)
∂θ
From (2.131) we obtain
2 θ
�
∂m
pr
= √
− rA θ = C = const,
(2.134)
∂θ �
1 + r � 2 + r 2 θ � 2
where C is identified (2.106, 2.134) as the conserved canonical angular momentum. In the case where C = 0, the ray intersects the
optic axis at some point. Such a ray has no angular momentum,
and is called a meridional ray. In the case where C 0, the ray
=
has angular momentum, and does not intersect the optic axis. Such
a ray is called a skew ray, with C as a measure of skewness.
53
2.5. Axial symmetry
�