48
Chapter 2. Geometrical optics
y(z) in the local transverse plane. Alternatively, one could treat
the local transverse plane as the complex plane with coordinates
u(z) = x(z) + iy(z) and u ¯(z) = x(z) − iy(z). Alternatively, one
could choose polar coordinates r(z) and θ(z) in the local transverse plane. The best choice is the one which allows one to express
the problem in the simplest possible way.
We can write
dx
n = P · ˆ s = P · ds
dx
dy
dz
= P x
+ P y
+ P z
ds
ds
ds
dz
= (P x x
� + P y y
� + P z ) ds
ds
m = n
= P x x
� + P y y
� + P z ,
(2.105)
dz
from which it follows
∂m = P j
(2.106)
∂x
�
j
for j = 1, 2, where P x and P y are the transverse components of
canonical momentum. The Euler-Lagrange equations can therefore
be written as
∂m dP j
−
= 0
(2.107)
∂x j
dt
in analogy with (2.58). Considering two rays which are infinitesimally displaced from one another, the differential in optical path
between the rays is (2.103, 2.106)
2
4
δW ab =
(P bj δx bj − P aj δx aj ).
(2.108)
j=1
In general δW ab is non-zero, since the endpoints x aj and x bj can be
independently displaced between the two rays by δx aj and δx bj ,
respectively.
Since δW ab is an exact differential, it follows that
∂W ab
∂W ab
P bj =
,
P aj = −
.
(2.109)
∂x bj
∂x aj