2.4.1 Equation of motion in terms of transverse coordinates and slopes
We found previously that the optical path length along a ray joining two endpoints x a and x b is given by the action integral (2.33)
as
x b
z b
W ab =
n ds =
m dz,
(2.98)
xa
za
where we have defined a modified refractive index m as
ds
m(x, x
� ; z) = n
= n 1 + x �2 + y �2 ,
(2.99)
dz
where x and x
� are the two-dimensional vector position and slope
components in the transverse plane, respectively, and where the
prime represents differentiation with respect to z. The variation of
optical path length is given by
z b
z b
δW ab = δ
m dz =
(δm) dz = 0.
(2.100)
za
za
46
Chapter 2. Geometrical optics
In fact, both the position and slope are considered small in the
following. Equivalently, we will only investigate rays that remain
close to the central axis.
Our purpose here is to identify the equations of motion, and describe a general methodology for solving them. This solution can
later be applied to a large variety of specific cases, describing a
similar variety of phenomena observed in practice. The reader is
referred to the references by Sturrock [86], Rose [75], Hawkes and
Kasper [43, 44, 45], and Wollnik [93] for further detail and elaboration. The present analysis is based on the earlier works of Glaser
[33] and Sturrock [86].
We found previously that the optical path length along a ray joining two endpoints x a and x b is given by the action integral (2.33)
as
x b
z b
W ab =
n ds =
m dz,
(2.98)
xa
za
where we have defined a modified refractive index m as
ds
m(x, x
� ; z) = n
= n 1 + x �2 + y �2 ,
(2.99)
dz
where x and x
� are the two-dimensional vector position and slope
components in the transverse plane, respectively, and where the
prime represents differentiation with respect to z. The variation of
optical path length is given by
z b
z b
δW ab = δ
m dz =
(δm) dz = 0.
(2.100)
za
za
46
Chapter 2. Geometrical optics
In fact, both the position and slope are considered small in the
following. Equivalently, we will only investigate rays that remain
close to the central axis.
Our purpose here is to identify the equations of motion, and describe a general methodology for solving them. This solution can
later be applied to a large variety of specific cases, describing a
similar variety of phenomena observed in practice. The reader is
referred to the references by Sturrock [86], Rose [75], Hawkes and
Kasper [43, 44, 45], and Wollnik [93] for further detail and elaboration. The present analysis is based on the earlier works of Glaser
[33] and Sturrock [86].