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�
54
Chapter 2. Geometrical optics
The Euler-Lagrange equation for the radial coordinate is (2.104)

∂m
d ∂m
−
= 0.
(2.135)
∂r
dz ∂r �
This leads (2.131, 2.135) to the exact ray equation for the radial
coordinate in the case of axial symmetry as follows:
⎡
⎤
d
� p 2 − (C/r + A θ ) 2
1 + r
� 2
1/2
⎣ r
⎦ =
dz
1 + r � 2
p 2 − (C/r + A θ ) 2
(C/r + A θ )
2
∂p
C
A θ ∂A θ
·
+ p
−
+ A θ
+
.
r
∂r
r
r
∂r
(2.136)
recalling that p is the scalar kinetic momentum (2.123), and A θ is
the magnitude of the magnetic vector potential (2.130). Both p
and A θ are assumed to be known functions of the coordinates.
The differential equations (2.134, 2.136) are a coupled pair, for
which the desired solutions r(z) and θ(z) are exact in principle.
These equations were derived by Sturrock [86]. Because the equations are nonlinear, the solutions r(z) and θ(z) cannot be expressed
in a simple, closed form. Consequently, an analytical solution must
rely on finding a suitable approximation. Alternatively, these equations are amenable to exact numerical solution.
2.5.2 Paraxial approximation, Gaussian optics
Assuming
r
� 2 « 1, C = 0
(2.137)